ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY NATHAN CANEN, MATTHEW O. JACKSON, AND FRANCESCO TREBBI Abstract. We develop a model of endogenous network formation as well as strategic interactions that take place on the resulting network, and use it to measure social complementarities in the legislative process. Our model allows for partisan bias and homophily in the formation of relationships, which then impact legislative output. We identify and structurally estimate our model using data on social and legislative efforts of members for each of the 105th- 110th U.S. Congresses (1997-2009). We find large network effects in the form of complementarities between the efforts of politicians, both within and across parties. Although partisanship and preference differences between parties are significant drivers of socializing in Congress, our empirical evidence paints a less polarized picture of the informal connections of members of Congress than typically emerges from congressional votes alone. Finally, we show that our formulation is useful for developing relevant counterfactuals, including the effect of political polarization on legislative activity (and how this effect can be reversed), and the impacts of networks in the congressional emergency response to the 2008-09 financial crisis. Date: February, 2020 Canen: Department of Economics, University of Houston. Mailing Address: 3623 Cullen Boulevard, Office 221-C, Houston, TX, 77204, USA. E-mail: [email protected]. Jackson: Department of Economics, Stanford University, CIFAR Fellow, and External Faculty of Santa Fe Institute. Mailing Address: 579 Serra Mall, Stanford, CA, 94305, USA. E-mail: [email protected]. Trebbi: Vancouver School of Economics University of British Columbia, and CIFAR Fellow. Mailing Address: 6000 Iona Drive, Vancouver, BC, V6T 1L4, Canada. E-mail: [email protected]. Kareem Carr and Juan Felipe Ria˜ no provided excellent assistance in the data collection. The authors would like to thank Matilde Bombardini, Gabriel Lopez-Moctezuma, and seminar participants at various institutions for their comments and suggestions. Particularly, we thank Antonio Cabrales and Yves Zenou for fruitful discussion of their model. Jackson gratefully acknowledges support by the Canadian Institute For Advanced Research and the NSF under grant SES-1629446. Trebbi gratefully acknowledges support by the Canadian Institute For Advanced Research and the Social Sciences and Humanities Research Council of Canada. A previous version of this work circulated under the title “Endogenous Network Formation in Congress”.
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ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
NATHAN CANEN, MATTHEW O. JACKSON, AND FRANCESCO TREBBI
Abstract. We develop a model of endogenous network formation as well asstrategic interactions that take place on the resulting network, and use it tomeasure social complementarities in the legislative process. Our model allowsfor partisan bias and homophily in the formation of relationships, which thenimpact legislative output. We identify and structurally estimate our modelusing data on social and legislative efforts of members for each of the 105th-110th U.S. Congresses (1997-2009). We find large network effects in the formof complementarities between the efforts of politicians, both within and acrossparties. Although partisanship and preference differences between parties aresignificant drivers of socializing in Congress, our empirical evidence paintsa less polarized picture of the informal connections of members of Congressthan typically emerges from congressional votes alone. Finally, we show thatour formulation is useful for developing relevant counterfactuals, including theeffect of political polarization on legislative activity (and how this effect can bereversed), and the impacts of networks in the congressional emergency responseto the 2008-09 financial crisis.
Date: February, 2020
Canen: Department of Economics, University of Houston.Mailing Address: 3623 Cullen Boulevard, Office 221-C, Houston, TX, 77204, USA.E-mail: [email protected].
Jackson: Department of Economics, Stanford University, CIFAR Fellow, and External Faculty of Santa FeInstitute.Mailing Address: 579 Serra Mall, Stanford, CA, 94305, USA.E-mail: [email protected].
Trebbi: Vancouver School of Economics University of British Columbia, and CIFAR Fellow.Mailing Address: 6000 Iona Drive, Vancouver, BC, V6T 1L4, Canada.E-mail: [email protected].
Kareem Carr and Juan Felipe Riano provided excellent assistance in the data collection. The authorswould like to thank Matilde Bombardini, Gabriel Lopez-Moctezuma, and seminar participants at variousinstitutions for their comments and suggestions. Particularly, we thank Antonio Cabrales and Yves Zenoufor fruitful discussion of their model. Jackson gratefully acknowledges support by the Canadian InstituteFor Advanced Research and the NSF under grant SES-1629446. Trebbi gratefully acknowledges support bythe Canadian Institute For Advanced Research and the Social Sciences and Humanities Research Councilof Canada. A previous version of this work circulated under the title “Endogenous Network Formation inCongress”.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 1
1. Introduction
Deliberative bodies, especially large ones, rely on informal interactions in order to function
productively. Individuals form relationships with each other to craft and pass legislation.
Because of the salient role of interpersonal ties in the legislative process, its study dates at
least to the 1930s (Routt, 1938), but only in the last fifteen years has this area of research
come to more prominence (Lazer, 2011).
The challenge of simultaneously modeling network formation and political decision-making
is one reason for this delayed uptake. Interpersonal ties are not drawn at random – legislators
strategically choose their connections (e.g., with whom to cooperate and collaborate). In
turn, then benefits are strategic as well – having key allies can enable a politician to craft
and pass legislation that would otherwise not be possible. Finally, these decisions are made
in an environment rife with identity-based (party) affiliation with an immense number of
possible connections, making empirical analysis challenging. The construction and analysis
of the model presented here addresses this challenge.
Our model simultaneously captures endogenous network formation, strategic decisions
made on this network, and homophily: group identity matters in payoffs and in link forma-
tion. We prove statistical identification of the parameters driving each of these features.1
In particular, we generalize the tractable and powerful framework of Cabrales, Calvo-
Armengol, and Zenou (2011) in several important directions. As in Cabrales et al. (2011), our
model has two strategic choices: legislators choose both how much socializing to do with other
politicians as well as how much effort to exert crafting and passing legislation. Socializing
efforts result in formed relationships that increase the success of legislative efforts, and so
social and legislative efforts are complements. Importantly, social and legislative efforts are
also complementary to those of the other politicians with whom a given politician has ties,
both within and outside of his/her party. The two main generalizations in our model are as
follows. First, while in Cabrales et al. (2011) relationships form completely at random, our
model admits homophily and allows social ties to form at a different rate within compared to
across groups – so that, for instance, legislators can collaborate with members of their own
party at a different rate than with members of the opposition. Second, we allow the returns
to social and legislative efforts to be party-specific. This captures important institutional or
time-specific differences across parties, including who holds a majority, which can make a
big difference in the returns to effort.
We structurally estimate our model employing data on cosponsorship and legislative efforts
of members of House of Representatives from the 105th-110th U.S. Congresses.
A first empirical finding is that the complementarities among politicians are significant
and stable across our sample period. The estimated social marginal multiplier on legislative
effort is between a tenth and a fourth of the direct incentive for legislative effort, with larger
1This model is the first to capture all of these features, which should be useful beyond the application tolegislative production. In Section 2.6, we compare our model to those in the literature.
2 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
values for Democrats.2 This means that a nontrivial fraction of incentives for efforts of
politicians appears to be driven by what other politicians are doing.
We then examine differences between Democrats and Republicans. We find that the two
parties have different base payoffs from passing legislation, both in terms of average and
variance (both higher for Democrats). These differences lead to higher levels of social and
legislative efforts by Democrats, all else held equal.
Further, we find evidence that partisan bias is an empirically relevant feature and the
model with biased interactions fits the data significantly better than a model with no bias.
We also show that our model that imputes endogenous network formation has better in-
sample properties (model fit) than models of social networks in Congress based on exogenous
or predetermined graphs, including those based solely on cosponsorships, alumni connections
and committee membership. As such, legislative behavior can be better explained by ex-
plicitly modeling the strategic decisions that drive these underlying networks, than using a
proxy for them.
We also show, however, that social interaction in the U.S. Congress is far from being an
exclusively partisan affair. The data are more nuanced than the common narrative of a
balkanized Congress, segregated along party lines, that has emerged from recent literature
mostly based on post-1980 congressional roll call evidence (Fiorina, 2017). We find that
intermediate levels of partisanship – a partisan bias is in the range of 5-15 percent – fit the
data significantly better than a fully partisan model where 100 percent of interactions are
exclusively within party. It is hard to reconcile the thousands of bipartisan cosponsorships
in recent data with a hypothesis of unmitigated polarization between parties. The stark
posturing and divisive language (Gentzkow and Shapiro, 2015; Gentzkow, 2017), and some
metrics of formal political activity, miss bipartisan interaction that more informally takes
place among legislators – especially with respect to the bulk of less controversial bills that
constitute day-to-day law and budget making.
We then assess the specific equilibrium within each Congress among the multiple equilibria
in this class of games. We show that the estimated equilibria are interior and stable, and
that social effort is (inefficiently) under-provided in all Congresses. There are externalities
in efforts, given the complementarities, that are not internalized by the agents.
Our model and structural approach provide comparative statics, which enable us to per-
form counterfactuals based on our estimated parameters. In a first illustration, we estimate
the effects of political polarization, such as a drifting to the right of Republican types, on
effort levels. We find that an increase of GOP ideological extremism of about 20% leads
to an average decrease of 8.3% in social effort for Democrats and a decrease of 7.5% for
Republicans. Further, we show that this decrease can be reversed by an improved selection
of legislators even by one party alone, as the higher legislative activity by that party can
2As it will be clear in the analysis that follows, the parameter multiplying the full product of social andlegislative efforts ranges from 0.03 to 0.05, which when multiplied by other legislators’ efforts of 4 to 6, andsocial efforts around 1, leads to a multiplier of 0.12 to 0.3. This is compared to direct incentives for legislativeactivity ranging from 1.0 to 1.4.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 3
spill over to the other. We also find that increases of bipartisanship (i.e., cross-party interac-
tions) do not imply unambiguous increases in governmental activity and legislative success.
A party can benefit from being less exposed to less-engaged low types in the opposition.
In another exercise, we examine how the amount of legislation that was generated in the
emergency response to the 2008-09 financial crisis would have changed if the Democrats had
not taken over the House in 2006. This emerged as a narrative for the evolution of the anti-
government intervention and Tea Party movement of 2010 (Mayer, 2016). Here we find a
quantitatively small change (between 1-2 percent lower likelihood of success) in the amount
of emergency legislation that would have passed had the legislature had remained identical
to the Republican 109th Congress, with virtually no difference in final outcomes.
1.1. Relation to the Literature. From the theoretical perspective, this paper contributes
to a literature that examines peer-influenced behavior when accounting for the endogeneity
of networks.3 Our model allows for homophily, so that people interact more within groups
than across groups, and also allows the value of social interaction to differ across groups.
This meaningfully generalizes the model of Cabrales et al. (2011) to have group membership
impact the value of social interaction and the rate at which that interaction happens within
compared to across groups. Both factors matter significantly in our empirical application.
Introducing a theoretically tractable and econometrically feasible form of asymmetry in the
process of socializing of members of Congress within this framework should be a valuable
base for other applications that involve multiple groups.4
This paper also contributes to the growing literature showing that social networks matter
in legislative environments. For instance, Fowler (2006) used a connectedness measure based
on cosponsorship to show that more connected members of Congress are able to get more
amendments approved and have more success on roll call votes on their sponsored bills.5 Also
using cosponsorship links, Cho and Fowler (2010) show that Congress appears subdivided in
multiple dense parts tied together by some intermediaries. These network features correlate
with legislative productivity over time (number of important laws passed, as defined by
Mayhew, 2005).6
3See Bramoulle, Djebbari, and Fortin (2009), Mauleon et al. (2010), Goldsmith-Pinkham and Imbens (2013a),Goldsmith-Pinkham and Imbens (2013b), Manski (2013), Jackson (2013), Badev (2017), Jackson (2019),Hsieh and Lee (2016), Mele (2017), Baumann (2017) (and see Jackson (2005), Jackson (2008), Jackson andZenou (2015) for surveys of the network formation and games on networks literatures).4Homophily in peer group formation is also theoretically explored in Baccara and Yariv, 2013, who furtherexplore group stability.5A similar study is Zhang et al. (2008).6There is other work on cosponsorhip. For example, Aleman and Calvo (2013), Koger (2003), and Brattonand Rouse (2011) study the incentives for cosponsoring in different settings (focusing on ideological similar-ity, tenure, etc.). Beyond their role in social networks, Wilson and Young (1997) study the signaling contentof cosponsorships, noting that cosponsorship is a cheap way of signaling to the median voter about one’s con-gressional activity. They identify three different explanations for cosponsorships and their possible signalingimpact: (i) bandwagoning (signaling strong support for the bill), (ii) ideology, and (iii) expertise. They finda null to moderate effect of cosponsorship on bill success, as measured as successive progress of the billsthrough Congress hurdles. Kessler and Krehbiel (1996) instead point out that the timing of cosponsorshipswould indicate that it is not as much a signaling to voters, as to other politicians (for example, they showthat extremists seem to cosponsor earlier). Still in the context of bill sponsorships, Anderson et al. (2003)
4 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
The network analysis of legislation is growing, and provides increasing evidence that social
relationships matter substantially and are causal in nature. For example, Kirkland (2011)
shows a correlation between bill survival and weak ties of the sponsor for eight state legisla-
tures and for the US House of Representatives. Cohen and Malloy (2014) employ identifica-
tion restrictions aimed at ascertaining causal effects of networks on voting behavior (using
the quasi-at-random seating arrangements of Freshman Senators).7 Rogowski and Sinclair
(2012) also use random spatial arrangements to estimate the causal effect of interactions
on legislative voting and cosponsorship in the House. In particular, the authors use lottery
office assignment affecting certain classes of members of the House of Representatives, not
finding a significant affect of office proximity on co-behavior.8 Harmon et al. (2019) studies
the role of exogenously shifted social connections within the European Parliament also using
seating arrangements.
Importantly, none of those papers model interaction as a choice variable. In an important
theoretical contribution, Squintani (2018) studies endogenous legislative networks, and the
role of ideological positions on information transmission. Our focus is on legislative produc-
tion rather than information transmission, and our analysis enables us to see how incentives
to socialize differ across parties, relate to legislative productivity, and have changed over time.
The role of political networks in connection to special interest politics is studied in Groll and
Prummer (2016) and Battaglini et al. (2018). In particular, Battaglini et al. (2018) focus
on legislative effectiveness of legislators based on their Bonacich centrality - taking it as ex-
ogenous, but employing a Heckman two-step procedure based on alumni networks to correct
for network endogeneity in the empirical analysis.9 In a complementary effort, we present a
tightly connected theoretical and empirical structure of endogenous network formation and
how it contributes to legislators’ decisions. This allows us to estimate socialization efforts
and how they affect legislation, and enables us to do comparative static/counterfactual ex-
ercises based on the estimated model. In subsequent work, Battaglini et al. (2019) study
a game of network formation with strategic decisions made on the graph in Congress. The
authors focus on the choice of directed links using a setting akin to general equilibrium:
while politicians can choose who to link to directly, in equilibrium the solution is charac-
terized by “prices” that clear the market for connections. We differ by modeling strategic
interactions within a game-theoretic framework and by proving identification (the focus in
find correlations with legislative productivity (i.e. the bill passing through different stages in Congress) forCongress member who sponsor more bills and use more floor time (albeit at a declining marginal rate).7For a similar approach see also Masket (2008).8They do not interpret these results as an absence of peer effects in Congress, but rather that office proximity- and the exogenous changes in connections caused by it - do not significantly explain congressional behavior.In fact, the most important network effects might be those from endogenously formed connections. However,these are hard to identify in reduced form and cannot be captured in their study. As they conclude, “Payingattention not only to the structure of networks but also to how that structure came to be can help remedymany of the difficulties in providing causal evidence for network effects.” (p. 327). Our structural model fillsthis gap.9For a similar reduced-form identification strategy, we exploit differential characteristics of networks withinthe same bill across Senate and House. The results are available from the authors upon request.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 5
the cited paper is on Bayesian estimation). We also conduct multiple policy counterfactuals,
which are precluded in that alternative framework.
2. The Model
2.1. Legislators, Parties, and Partisanship. The legislature, henceforth referred to as
“Congress”, is composed of a set N = 1, 2, ..., n of members. For simplicity, we focus
on one chamber (e.g. the House), and clearly although we refer to Congress, the model
applies to a variety of deliberative bodies, legislatures, committees, and organizations more
generally.
The set of politicians N is partitioned into parties, with a generic party denoted P`.
Each party P` has a level of partisanship p`. This can be thought of as a structural
form of homophily. In particular, members of party ` spend a fraction of their interaction,
p`, at exclusively party ` events, so only mixing and meeting with own-party events, and
the remainder, 1 − p`, at events in which they mix with members of all parties. This
can include party and caucus meetings, joint sessions, fund-raising events, committee works,
social gatherings and formal events, etc. For our period of interest, examples of party-specific
events are closed sessions called Party Conferences for Republicans and Party Caucuses for
Democrats (their respective chairs represent the number 3 position of official party leadership
rankings).
Politician i is from party P (i), and p(i) denotes the level of partisanship of politician i’s
party.
In our empirical analysis, there are two parties, 1, 2, and then we index the n politicians
so that the first q of them belong to P1 = 1; ...; q and the remainder to P2 = q + 1; ...;n.Let q ≥ n/2, so that party 1 is the majority party.
2.1.1. Socializing. Each politician chooses an effort level of how much he or she socializes
(i.e., interacting with other politicians), denoted si ∈ <+. It is via this socializing that he
or she forms connections with other politicians.
The network G = gi,ji,j∈N that arises from the vector of social efforts, s, is described
by10:
gij(s) = sisjmij(s),
where if j ∈ P (i) then
mij(s) = p(i)p(j)∑
k∈P (i),k 6=i p(k)sk+ (1− p(i)) (1− p(j))∑
k 6=i(1− p(k))sk,
10This is for the case in which sj > 0 for at least two people in each party. If other agents are not putting insocial effort, then there can be nobody to match with, and then some of these equations do not apply (theydivide by 0, and if all s are 0). In those cases the matching is described as follows. If at most one sj > 0,then set mij = 0 for all ij and the entire network equal to 0. If there are at least two people with sj > 0,but also at least one party with sj > 0 for no more than one agent, then set mij = gi,j = 0 for all membersof a party that has nor more than one sj > 0, but use the remaining above specified equations for gi,j ’s forany other combinations.
6 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
and if j /∈ P (i) then
mij(s) = (1− p(i)) (1− p(j))∑k 6=i(1− p(k))sk
.
So, politicians meet their own party members in two ways: at their own events and at
bipartisan events. They meet members of the other party only at the bipartisan events.
Politicians are met with the relative frequency with which they are present at events.
This specification captures key features of how politicians socialize in practice, while main-
taining tractability of the model. First, members of the same party meet more often and,
hence, are more likely to connect with one another.11 Second, more social members (those
with higher si, which we will show to be the higher types in equilibrium) are more likely to
connect to members in their own party.12 Third, socialization is not deterministic: observ-
able decisions do not fully determine social connections. In this model, two socially active
members with equal characteristics will not necessarily link. As in any activity, we allow for
randomness to be a component of social relationships, although driven by efforts and social-
ization characteristics. In practice, personalities, unobserved characteristics and preferences
play a large role in such connections. Furthermore, the events in which legislators interact
are often unobserved by researchers and the public.13
Our specification with a biased random matching protocol allows us to capture both these
realistic socialization features, as well as maintaining a tractable framework in which we
can characterize equilibria, obtain theoretical predictions, and test them empirically. As an
alternative one could consider a model of directed links gij, where politician i identifies a
specific partner j deterministically and based on j’s other choices. This appears less tractable
and unrealistic. Not only is the action space gigantic (with 2435 potential directed links just
in the network formation stage), but these links need to endogenously anticipate all pairs’
strategic decisions over the entire network. Our framework instead allows for closed-form
solutions and for a realistic modicum of stochasticity in the formation of certain social ties.
In Appendix A we show that: ∑j 6=i
sisjmij(s) = si,
so that the total number of connections that i makes is proportional relative to si.
When p` = 0 for each `, then this simplifies to coincide with the model of Cabrales et al.
(2011). When p` = 1 for each `, instead, each party is completely cut off from the other.
Then, within each party again Cabrales et al. (2011) applies.14
11As previously described, examples of single party events include fund-raisers and party caucuses.12We will explore these patterns empirically in a later section, showing that these higher types coincide withthose in more influential committees, for example.13These include being at the gym at the same time, going to particular restaurants or attending certain cul-tural events. A journalistic description for the case of the Senate can be found in Roll Call’s piece, https://www.rollcall.com/news/behind_the_doors_of_the_senate_gym-222790-1.html, accessed January 21,2019.14One could also consider endogenizing partisanship. Having three action variables for each party/agentwould render the model intractable analytically. Here we focus on the two that seem most important to
2.2. Legislative effort and Preferences. The other choice of politicians is their legislative
effort xi ∈ <+. The benefits from legislative efforts are described by:
αixi + φi∑j 6=i
sisjmij(s)xixj.
As in a large class of models, of which Cabrales et al. (2011) is a salient instance, there
is a direct benefit from private effort, with idiosyncratic weight αi. In addition, there are
complementarities in legislative efforts between politicians who have formed connections:
the more effort they both expend, the more likely their legislation is to pass. The size of
this interaction effect is governed by a party-specific parameter φi, whose quantification is
a relevant goal in the empirical analysis that follows. Note that φi = φP (i), but we keep the
first notation for simplicity.
Both forms of effort are costly for a politician. The cost of legislative effort is given byc2x2i , with c > 0, and the total cost of socializing is given by 1
2s2i . The parameter c governs
the relative cost of legislative effort to social effort.
Taken together, the politician’s preferences are the amount of legislation that he or she
produces less the costs of legislative and social efforts. This is given by:
(2.1) ui(xi, x−i, si, s−i) = αixi + φi∑j 6=i
sisjmij(s)xixj −c
2x2i −
1
2s2i .
If G was exogenous and known, equation (2.1) would collapse to the set-up in Ballester
et al. (2006). As a result, equilibrium legislative efforts would be proportional to a weighted
measure of the politician’s centrality. If that were the case, we could base an empirical
specification on this foundation, as done by Acemoglu et al. (2015) in the context of public
good provision on a geographical network. However, in this paper, we assume that the
politician network is endogenous and unknown. As a result, we must model the choice of
social effort that generates the underlying network in a way that can be identified in the
data. The current set-up accomplishes this through equilibrium restrictions and additional
data in an appropriate way, as we show in the next sections.
2.3. A Micro-Foundation Built Upon Reelection Preferences. There are many dif-
ferent ways in which we could justify the preferences in (2.1), as the natural presumption is
that politicians care to maximize the legislation that they pass. Here, we posit a realistic
microfoundation, which we will bring directly to the data.
Politicians care about being reelected and can affect the probability of being reelected
by exerting effort in Congress and by building connections instrumental to having specific
legislation passed (e.g. policy favorable to the politician’s constituents).
Each politician anticipates these effects on his/her reelection chances. More specifically,
each congressional cycle has two periods, 1 and 2, where the second period provides the
reelection incentives that drive activity in the first period. Politicians are career motivated
and exert costly efforts with the aim of increasing their chances of being reelected.
endogenize, and then estimate the third. At the end of Appendix B, we discuss some potential approachesto, and challenges with, endogenizing partisanship that could be addressed in future research.
8 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
In period 1, each Congress member can present a policy proposal, which for brevity we
refer to as a “bill”. The bill consists of a policy goal the Congress member intends to
fulfill, for instance passing a statute targeted to his or her constituency, landing a subsidy,
or obtaining an earmark beneficial to firms in the home district. We describe below how
getting i’s policy goal fulfilled maps into an increase in i’s chances of being reelected.
Suppose a politician’s utility is given by:
(2.2) ui = Pr(reelected)− c
2x2i −
1
2s2i .
The choice of xi, the level of legislative activity exerted by i, affects the support for i’s
legislation, Yi, through a function:
Yi = εixi
(∑j∈N
gi,j(s)xj
).
Both i’s own legislative effort, xi, and that of his or her connections in the network,∑
j∈N gi,j(s)xj,
matter for the ultimate support received by i’s bill.
Yi is stochastic and depends also on a random shock εi, assumed to be standard Pareto
distributed with scale parameter γP (i) > 0 and i.i.d. across politicians. We allow γ to be
party-specific, reflecting different average electoral returns by party. We assume that εi is
realized after the choice of x, the vector of xj across all politicians j ∈ N . Because εi is a
shock following the realized legislative support, i must take expectations over its value when
choosing (xi, si).15
The bill is approved if Yi > m, where m > 0 is a generic institutional threshold.16 The
probability of having the bill approved is thus given by:
Pr(Yi > m) = Pr
εi > m
xi
(∑j∈N gi,j(s)xj
)(2.3)
=(γP (i)
m
)(∑j∈N
gi,j(s)xj
)xi,
where we use the distributional assumption on ε.17 Actual passage of the bill sponsored by
i is represented by the indicator function I[Yi>m].
We interpret xi as the observable legislative effort by i, instrumental to the approval of i’s
bill, and we postulate that voters prefer politicians exerting higher legislative effort to lower
effort. We also allow for voters to care about whether in fact the bill passes conditional on
15Also notice that each link between politician i and j is an endogenous function of the social efforts ofeverybody else, hence the dependency gi,j(s) on s, the vector of sj efforts across all politicians j ∈ N .16Naturally, m can be function of a simple majority requirement or even supermajority restrictions.17More generally, one can take Yi to represent the average approval rate of i’s multiple bills. In this case,each b is a separate bill by a politician i. The conditions for our model are unchanged, as long as bills arenot strategically introduced (i.e., the shocks εb are still i.i.d. within i).
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 9
effort. That is, we allow for the political principals (the voters) to reward their agent i for
effort xi, networking si, and ultimately luck εi.
To get reelected, the politician must have an approval rate in his/her electoral district that
is sufficiently large. Similarly to Bartels (1993), the electoral approval rate of i is modeled
as a variable Vi:
Vi = ρVi,0 + ζP (i)I[Yi>m] + αixi + ηi
where ηi is assumed to be a mean zero electoral shock, uniformly distributed on [−0.5, 0.5],
and where Vi,0 ≥ 0 stands for the baseline approval rate before the start of the term (i.e.
before period 1 in the model). Hence, this set-up allows for approval rates to be persistent,
but also to react when a politician is capable of getting a bill approved I[Yi>m] or when i
exerts high legislative effort xi. The parameter ζ, which could be equal to zero empirically,
governs the relative importance of a bill actually passing vis-a-vis legislative effort. The
direct effect of xi is captured by αi, which is i-specific and may depend on party affiliation,
majority status, congressional delegation, etc. Finally, notice that, while there is no direct
value to the voters of the politician having more socializing, the value of si matters implicitly,
being instrumental in getting legislation approved.
In period 2, i is reelected if his/her electoral approval level, Vi is larger than an electoral
threshold w < 1. So, the probability of being reelected is given by:
m. Replacing (2.4) into the utility function (2.2) yields:
ui(xi, x−i) = (1.5− w) + ρVi,0 + φi∑j∈N
gi,j(s)xixj + αixi −c
2x2i −
1
2s2i
18The above expression is non negative, since its terms are non negative.19When the probability of reelection is either 0 or 1, we have that si = xi = 0. This can be seen fromequation (2.2). If a politician i cannot influence his reelection prospects, he will not undertake costly effort.This is contrary to what is observed in the data, as described in Section 3, as well as theories of legislativebehavior (e.g. Mayhew (1974)).
10 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Since the terms (1.5− w) and ρVi,0 do not affect the maximization problem, (2.3) can be
rewritten as the specification given in (2.1).
2.4. Solving For Equilibrium. We examine the pure strategy Nash equilibria of the game
in which all politicians simultaneously choose si and xi.
The first order conditions with respect to si and xi that characterize the best response of
politician i imply that interior equilibrium levels of (s∗i , x∗i ) must satisfy:20
(2.5) s∗i = φi∑j 6=i
s∗jmij(s∗)x∗ix
∗j
and
(2.6) cx∗i = αi + φi∑j 6=i
s∗i s∗jmij(s
∗)x∗j .
We rewrite (2.5) as
(2.7)s∗ix∗i
= φi∑j 6=i
s∗jmij(s∗)x∗j ,
To fully characterize equilibria, we work with the same approximation as in Cabrales et al.
(2011). We operate “at the limit”, when the number of politicians grows.21 In particular,
we solve for equilibrium under the assumption that∑
j 6=i s∗jmij(s
∗)x∗j is the same for all i of
the same party.
This implies thats∗ix∗i
is the same for all agents within a party. Using (2.7) in (2.6) yields:
cx∗i = αi + s∗iφi∑j 6=i
s∗jmij(s∗)x∗j
= αi +s∗2ix∗i.
Dividing through by x∗i implies that
(2.8) c =αix∗i
+s∗2ix∗2i
.
Sinces∗ix∗i
is the same for all agents within a party provided that φi is the same for all agents
within a party, (2.8) implies that αix∗i
is the same for all agents within a party. This further
implies that:
x∗i = αiXP (i),
for some XP (i). In addition, the fact thats∗ix∗i
is the same for all agents within a party,
implies that
s∗i = αiSP (i),
20Note that second derivatives are everywhere negative.21Alternatively, this could be justified via a continuum of politicians of each type, or by examining an epsilonequilibrium with a large n.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 11
in equilibrium for some SP (i).22
To get explicit expressions for our empirical analysis of Congress, we now specialize the
analysis to the case of two parties.
For each party j = 1, 2 define
Aj =∑i∈Pj
αi,
Bj =∑i∈Pj
α2i .
Proposition 2.1. The (interior) Nash equilibria of the limit game of this model are positive
solutions to the system given by:
x∗i = αiXP (i), and(2.9)
s∗i = αiSP (i),(2.10)
where
(2.11)S1
X1
= φ1
(p1B1X1
A1
+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2
(1− p1)A1S1 + (1− p2)A2S2
),
(2.12)S2
X2
= φ2
(p2B2X2
A2
+(1− p2)2B2S2X2 + (1− p1)(1− p2)B1S1X1
(1− p1)A1S1 + (1− p2)A2S2
),
(2.13) cX21 = X1 + S2
1 , cX22 = X2 + S2
2 .
All proofs appear in Appendix A.
If p1 = 1 or p2 = 1, then things reduce to the case of two separate parties with no
interaction across them. That is, they are two copies of the model in Cabrales et al. (2011).
Similarly, if p1 = p2 = 0 then there is no impact of party affiliation, and again the model
simplifies to that of Cabrales et al. (2011). The novel case is when at least one partisanship
level is positive, yet both levels are below 1. This biases the interaction of at least one party,
leaving room for interaction across parties. In this case there will be both social mixing
across different parties and partisanship in socializing.
Generally, there are multiple equilibria. For instance, there is always an (unstable) equi-
librium in which si = 0 for all i. In that case, since no other politician provides effort, a
given politician’s efforts results in no connections and so the best response is also to provide
no effort.
A sufficient condition for existence of an interior equilibrium is as follows.
22In contrast to results in network games with exogenous networks, equilibrium actions in our model are notexpressed as only being proportional to a centrality measure of the network (e.g., Katz-Bonacich centrality).This results from the equilibrium interactions between social and legislative efforts. The legislative effortsstill have to satsify a version of the usual characterization on the margin. Therefore, an empirical approachusing only centrality measures for estimation instead of our structural equations would only capture onedimension of the model’s predictions.
12 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Proposition 2.2. A sufficient condition for the existence of an interior equilibrium is
2c3/2
3√
3≥ max [φ1, φ2] max
[B1
A1
,B2
A2
].
In this setting with two parties and nontrivial partisanship, there will generally be either
two or four interior equilibria (except at a degenerate set of values where the system switches
from two to four equilibria).23
2.5. Pareto Efficient Efforts. Before proceeding with the empirical analysis, we comment
on the Pareto inefficiency of the equilibrium outcomes of the model. This is relevant for a
welfare analysis that checks whether there is over-provision or under-provision of social and
legislative effort in the strategic setting.
Generally, the fact that there are positive externalities in efforts – in particular in legislative
efforts – implies that there is under-provision of effort. In particular, the Pareto optimal social
and legislative effort levels are unbounded: any finite level of efforts are Pareto dominated
by some higher levels.24 Hence, all equilibria are characterized by an “under-provision” of
efforts.
To see this, we first note that the interaction term in equation (2.3) multiplies three
variables together: si, xi and xj. This has a cubic function property on efforts: doubling all
efforts produces an eight times higher interaction effect. Meanwhile, the costs on the social
effort (si) and legislative effort (xi) are quadratic. Hence, doubling those only quadruples
costs. It is then direct to check that the gains from increasing effort grow faster than their
costs, which implies the following result.
Proposition 2.3. Every finite profile of efforts is Pareto dominated by some larger level of
efforts.
This implies that, although higher payoffs are possible, the selfish attention to individual
costs limits the amount of effort that is produced in any equilibrium.25 If we were to cap
effort levels at some high level, then there would exist Pareto optimal efforts bound by the
caps.
23These equilibria correspond to when both parties exert high levels or low levels of social efforts, and thenfor some parameters there are also two additional equilibria in which one party does medium-high and theother does medium-low socializing.24The Pareto analysis in Cabrales et al. (2011) only applies if actions are bounded at some small enoughfinite level. The second derivatives in their proof flip signs if actions are large enough. Thus, there is alocal maximum of a weighted sum of utilities that is interior (which is the one identified in their analysis offirst-best actions), but the global maximum is actually unbounded. With a strict enough bound on efforts,there would exist an interior maximizer. Effectively, once the efforts are large enough, then the interactioneffects dominate the costs. One needs to constrain efforts to be below that level in order to get an Paretooptimal effort solutions. Note however that even with bounds, the equilibrium efforts tend to be inefficient,given the externalities.25It should be noted, that with high enough effort levels, then the best responses increase without bound inresponse to increases in others’ efforts. There is no equilibrium, because of the unbounded feedback. Again,if one imposed a cap, then there would be an equilibrium at those capped levels in the model.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 13
The message here is that generally, given the complementarities and positive externalities,
there is underprovision of effort. A political party, or a government, could help overcome
some of the inefficiencies, for instance, by subsidizing meetings and interactions.
2.6. Discussion of the Model.
The model is designed to include five key features simultaneously: (i) strategic decisions on
a network - agents choose behaviors and their payoffs from those behaviors depend on their
neighbors’ behaviors, (ii) endogenous network formation - agents have discretion over whom
they interact with, and base those choices over the anticipated payoffs from their network
position accounting for the ensuing behaviors and externalities, (iii) group memberships and
homophily - both the payoffs and meeting rates of agents can be biased to privilege group
identity, (iv) statistical identification of the three previously discussed different features,
and (v) practical estimation of this model for a moderately sized network from a single
observation of a network and associated behaviors.26
These five features are all very important for our context and we anticipate they would
be in many other applications. This list of desirable features of the model requires some
stylizing of the model on various dimensions, trying to hit an appropriate point in the
tradeoff between tractability and richness. The model has to be rich enough to provide a
good fit and explain much of the variation in the data along several simultaneous dimensions
(more on this below), and yet be tractable enough to solve and estimate.
There are models that combine one or more of the various features mentioned above,
but none that allow for all of them. The growing literature on games on networks (e.g.,
see Jackson and Zenou (2015) for a survey) addresses peer interactions on networks and
how those vary as a function of the network. The linear-quadratic framework here, first
explored in Ballester, Calvo-Armengol, and Zenou (2006) has become a standard approach
in that literature given its tractability. Here this is combined with network formation and
homophily.
The extensive literature on network formation, starting from its early incarnation in Jack-
son and Wolinsky (1996); Dutta and Mutuswami (1997); Bala and Goyal (2000); Currarini
and Morelli (2000); Jackson and Watts (2002); Jackson (2005); Herings, Mauleon, and Van-
netelbosch (2009) provides insight into how networks form, when inefficient networks form,
and how that depends on the setting. More recently, the literature has also begun to de-
velop models that incorporate some heterogeneity and are still tractable enough to allow for
fitting the models to data, as in Leung (2015); Sheng (2016); Chandrasekhar and Jackson
(2016); Mele (2017); Graham (2017); de Paula, Richards-Shubik, and Tamer (2018); Leung
(2019); and some of that literature also allows for homophily, such as Currarini, Jackson,
and Pin (2009, 2010); Banerjee, Chandrasekhar, Duflo, and Jackson (2018). The models
that are tractable enough to fit to data require a structure that limits the multiplicity of
stable (equilibrium) networks, and such that those can be estimated with a practical number
of calculations.
26Congressional data allow for a time series, but the agents and their preferences change over time, and sowe need a model that can be fit from a snapshot of the network and behaviors.
14 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
The breadth of that class of models is still unknown and we only have a handful of such
estimable models that involve non-trivial network effects (payoffs that are not simply inde-
pendently determined link-by-link); and generally those models are stylized in some way.
For instance, as shown in Sheng (2016), the choice of the specific model can be important
as some models with indirect network effects (utility from friends-of-friends) lead to (i) a
lack of identification (many configurations of parameters leading to the same outcomes) and,
(ii) computational intractability with as few as 20 players, due to a curse of dimensional-
ity (p. 14). To make progress, she restricts the model to one with endogenous links that
have “dependence [that] has a particular structure such that conditional on some network
heterogeneity and individual heterogeneity, the links become independent.” An alternative
approach is that of Mele (2017). He proposes an empirical model of network formation that
allows for homophily in network formation. Again, he shows that there is a curse of dimen-
sionality in using standard estimation methods unless some strong asymptotic independence
conditions are satisfied. Other approaches are to have certain subgraphs generate value and
then model the formation of those subgraphs directly (Chandrasekhar and Jackson, 2016),
or to have payoffs based on combinations of individual characteristics, geography, or assor-
tativity (e.g., Currarini et al. (2009); Leung (2015); Graham (2017); Leung (2019)). Here
we want a model in which the value to a given pairing depends on their subsequent mutual
(legislative) efforts, and so we need a model in which expected values of links can be easily
calculated conditional upon future efforts, and those efforts can also be characterized easily
as a function of the pairings. Using random meeting probabilities to derive link formation
does exactly this by reducing the dimension of choices while allowing for interdependencies
and still yielding a clean characterization of both types of efforts.
In summary, one has to be judicious in modeling network formation to obtain a formulation
that is both well-identified and estimable, and to have heterogeneity. Meanwhile, existing
empirical models of games on networks that are well-identified (e.g., de Paula et al. (2019),
advancing the work of Bramoulle et al. (2009)) do not allow for endogenous networks - they
assume that the network is fixed and exogenous, and require a different data set-up than
ours.27
Thus one can see why models that incorporate both behavior and network formation
are few: Cabrales, Calvo-Armengol, and Zenou (2011); Konig, Tessone, and Zenou (2009);
Goldsmith-Pinkham and Imbens (2013b); Hiller (2017); Badev (2017). These models neces-
sarily sacrifice some richness in order to incorporate both network formation and endogenous
behaviors and to allow for an interaction between them. Nonetheless, they can still be quite
rich and, as we show here, can still fit data well. Of this class, in order to work with a
tractable model that we can extend to have yet a third dimension of group identity and
27For example, to recover an unobserved exogenous network as they set out to, de Paula et al. (2019) assumes(i) an exogenous network that is sufficiently sparse, (ii) the network does not change over time, (iii) a paneldata structure, with large enough time-series dimension, and (iv) a linear in means model. Our set-up anddata structure do not have any of these 4 properties, as alluded to previously. Furthermore, the estimationof this set-up must involve shrinkage estimators and their resulting bias due to the size of the parameterspace (N2 parameters to recover from just the network itself).
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 15
homophily, and still take to (static) data, we build upon the model of Cabrales, Calvo-
Armengol, and Zenou (2011). This introduces another dimension to the estimation, of group
interaction rates, and thus requires that the model be tractable enough to still solve with a
third dimension of endogeneity. Finally, a close look at the fit of the data provides support
for our modeling choices.28 In Section 5.1, we show that the predicted links from the model
are highly correlated with measures that use disaggregate data between congressmembers
(e.g. Fowler (2006)), even if we do not use the latter in estimation. Secondly, strategic
outcomes on this network, such as the probability of bill approval, are fit well within and
across parties/politicians. Third, homophily is quantitatively important: a model that with
homophily fits the data significantly better than one without it. Fourth, our model is shown
to outperform alternative approaches to the characterization of G in terms of in-sample mean
squared error. Finally, this model allows us to investigate behavior beyond what is observed:
it allows us to perform meaningful counterfactuals, shown later in the paper. Such exercises
include the effects and reversal of polarization, and the impacts of networks on the passage
of key bills in the Great Recession.
2.7. Preliminaries to Estimation. Generally, effort levels s∗i , x∗i are not measured exactly
and are observed with noise. For instance, the bill cosponsorships often used as the basis for
the construction of political networks are end products that miss other forms of socializing
(e.g. close-doors meetings, fund-raisers, and so on). Similarly, although we can partially
observe legislative effort through standard proxies (e.g. times the Congress member was
present on the floor for speeches, presence in roll call voting, or number of bills written29),
these are imperfect proxies for the legislative efforts that politicians exert. Thus, we account
for measurement error in our analysis.
LetNτ denote the politicians comprising Congress τ and note that this is a set which varies
across different τ .30 Introducing classical measurement error, for politician i in Congress τ ,
we observe:
si,τ = s∗i,τe−εi,τ(2.14)
xi,τ = x∗i,τe−vi,τ .(2.15)
s∗i denotes what is chosen, but it is hit with independent noise and si is observed (and
similarly for xi). The measurement error, conditional on this observation, is mean zero, and
independent of all the other measurement errors across individuals and time. We do not need
to impose that the measurement errors in both types of effort have the same distribution.
28Most notably, we see the value of (i) a (biased) random socialization protocol, (ii) choices made on effortlevels, and (iii) mean-zero i.i.d. measurement errors on our observed proxies.29Both highlighted as important for legislative success in Anderson et al. (2003).30The data is observed for multiple Congresses and we provide identification results for parameters specificto each Congress. This means we allow our parameters to differ across different Congresses and we canconstruct time-series estimates of the parameters.
16 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
From Proposition 2.1, (S1, S2, X1, X2) are completely determined by the parameters that
govern the system. Then, all individual choices are functions of parameters and of the set of
types αjj∈Nτ .Let:
(2.16) αi,τ = ez′i,τβP (i),τ
where zi,τ indicates a vector of individual observables31 (e.g. ideology, tenure, committee
membership), and βP (i),τ are party-specific and Congress-specific parameters that will be
estimated.32
Measurement error’s independence with respect to individual politician covariates is not
a particularly stringent assumption in our context and is needed to allow the possibility of
a partial mismatch between data and model effort predictions. Covariates capture much of
the individual-level heterogeneity in benefits from legislative effort and common behavior:
in the model, through αi, and in the data, as we allow αi to vary according to ideology,
tenure, strength of committee positions held (this is verified in the model fit section). As
a result, our i.i.d. assumption is on the (mis)measurement of equilibrium actions, not the
actions themselves. It implies that our mismeasurement on average is not worse for certain
politicians than others, conditional on their characteristics. While we make this assumption
explicitly in this paper, it is commonly (implicitly) used for inference in most empirical
models, whether reduced form (e.g. using instrumental variables as in Battaglini et al.,
2018) or structural for the validity of normal approximations of test statistics/asymptotic
distributions.
In addition, our specification is not oblivious to common shocks driving social and leg-
islative effort and legislative success. Quite the contrary, Proposition 2.1 shows how our
structural approach in fact operates under the theoretical result of dependence of individual
efforts from party and time specific common XP (i) and SP (i) factors through Equations (2.9)-
(2.10). The model’s system of equations makes explicit how common and possibly correlated
shifts affecting all members of a specific party in a Congress affect individual equilibrium
choices. In fact, XP (i) and SP (i) work as sufficient statistics that capture all common be-
haviors across the network - so only individual characteristics and those parameters must
be specified to fully capture the endogenety of network formation. In contrast, an approach
based on Ballester et al. (2006) would have to specify a full, observable and exogenous,
network and deal with its nonlinearities. We revisit this after the Results section, when
discussing model fit.
31Unobservables are already present when we introduce measurement errors. Note that if we had αi =ez
′iβ+ηi , we could rewrite equation (2.14), using the equilibrium results, simply as:
sieεi−ηi = ez
′iβSP (i),
and a redefinition of the measurement error to εi − ηi (still mean-zero and i.i.d.) would suffice in returningto the model presented in the main text, as long as A1, A2, B1, B2 were not functions of ηi.32Identification of the model does not rely on the parametrization of α, as we prove in Appendix I. However,this is useful for estimation purposes. A nonparametric α would require us to estimate a parameter αi foreach politician in each Congress, when we only observe one set of (si, xi) per period.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 17
The information we employ in the analysis is the following. Let yi,τ = I(Yi,τ>m) indicate
whether each bill was approved or not, where i ∈ Nτ and τ is a given Congressional cycle.33
si,τ indicates the (log of hundreds of) cosponsorship decisions per politician i ∈ Nτ . This
is our proxy for the equilibrium social efforts∗i,τ
. The use of logs and rescaling allows us
to keep this effort proxy in the same scale as our proxy variable for legislative effort. xi,τindicates a vector of observable proxies for legislative effort
x∗i,τ
. As discussed in more
detail in the following section, this is constructed using data on floor speeches (word counts
per politician during a term) and roll call presence/votes. We employ a procedure (Non-
Negative Matrix Factorization, see Trebbi and Weese, 2019), to reduce the dimensionality
of this set of proxies to a single dimension.34
As we perform our analysis within a Congress, we suppress the notation τ . We assume
that a single pure strategy Nash equilibrium, as defined in Proposition 2.1, is played in each
Congress. We do not impose, however, that the same equilibrium is played across different
Congresses, rather we characterize the equilibrium played empirically in Section 6.
Given yi, si, xi, zii∈N , we estimate the parameters (c, φ1, φ2, ζ1, ζ2, γ, p1, p2, β1, β2). For
identification, we set m = 1, so that the random variable εi is scaled in terms of the institu-
tional threshold. The basis for identification is Proposition 2.1 and the systems of equations
that it provides.
For identification of the parameters of our model it is not necessary to identify the full set
of equilibria, but instead just to use the implications that we are observing some (interior)
equilibrium. More precisely, we show that, given the observed data, one can uniquely pin
down the equilibrium that is played, as long as only one is played during each Congressional
term. Formal identification of our model is demonstrated in Appendix D.
3. Data
We use the cosponsorship data from Fowler (2006), compiled from the Library of Congress,
covering the 105th to the 110th United States Congress (from 1997 to 2009). This data
contains cosponsorship decisions by politician, and within that data, who sponsors and
who cosponsors each bill. It also contains information on whether each bill was approved in
Congress or not (we focus on passage in the House of Representatives). Figure 1a shows that
measures of inter-connectedness of Congress, for example the total number of cosponsorship
links in legislative acts across members of the House (Fowler (2006)), have been steadily
increasing. Figure 1b then breaks down how cosponsorships vary within and across parties.
Per Congressional cycle, we compute the log of how many hundred bills each politician
cosponsors, which is the variable Cosponsorships. This function of cosponsorships acts as an
empirical proxy for the social effort s∗i,τi∈Nτ .We note here that cosponsorship differs from bill sponsorship. Sponsoring a bill refers to
the introduction of a bill for consideration (and can be done by multiple legislators drafting
the bill, the “sponsors”). These sponsors are the authors of the bill. Instead, cosponsorships
33A complete data description section follows below.34Sponsorship of bills is already included, as we use the separate bills independently. Further details on this,the data and procedure to lead to the effort proxy are in the next section.
18 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Figure 1. Number of Cosponsorships per Congressional Cycle
(a) Total Number of Cosponsorships
(b) Number of Cosponsorships Within and Across Parties
The figure shows the evolution of the total number of (unique) cosponsorships during a congressionalcycle (i.e. anytime a politician has cosponsored another in a directed way) over time. The first figureshows the total number of cosponsorships, while the second decomposes it by party.
refer to the decision of adding one’s name as a supporter of the bill (becoming a “cosponsor”
of the bill). In contrast to sponsorship of a bill, the decision to cosponsor does not involve any
writing of legislation. Instead, cosponsorships serve as a signal of support to that current bill
(or potentially, to its authors), without ownership of the legislation itself. Cosponsorships
are prevalent in Congress, as can be seen in Table 1, and the presence of cosponsorship
across party lines is still quite common, notwithstanding the trends in polarization discussed
in Fiorina (2017) or Canen et al. (2020), as evident from the time series in Figure 1b.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 19
The individual bill success outcome (i.e. if the bill passes or not) maps into yi,τi∈N .
We then use the sponsorship information to link the outcome of the bill to the network
characteristics and individual decisions.
To compute our proxies for legislative effort, x∗i,τi∈Nτ , we first collect data on Roll Call
voting and floor speeches in Congress. Data for Roll Call voting comes from VoteView.
We compute an index, for each politician and for each term in Congress, as the times the
Congress member voted as a proportion of total Roll Call votes. This measure, which we
call Roll Call Effort, is defined as 1− (number of times i was “Not Voting”/ total Number
of Roll Call votes in a Congress).
Following Anderson et al. (2003), we also use data on floor speeches as a measure of
individual legislative effort. To do so, we compile the amount of words that each Congress
member used in his/her floor speeches across the duration of one term. Our Floor Speeches
variable is constructed as log(1 +Wordsi,τ/100). We log and rescale this variable to a scale
comparable to other legislative activities.35 Data on floor speeches comes from Gentzkow
and Shapiro (2015), available on ICPSR.36
That these measures of social interaction and legislative activity may be germane to one
another is evident from the significant and positive raw correlation of link formation and
proxies of legislative activity and effort, for instance floor speeches in Figure 2. This com-
plementarity between effort choices is fully consistent with our theoretical setup.
We proceed to construct xi,τi∈Nτ , by using both Roll Call Effort and Floor Speeches. An
appropriate combination of these variables can be obtained through dimensionality reduc-
tion methods. Since effort should be non-negative, we employ a procedure that guarantees
positive values (i.e. we do not use methodologies like principal components analysis that
involve a centering of data and negative values).37 We employ Non-Negative Matrix Fac-
torization (NNMF), a dimensionality reduction procedure which imposes constraints so that
the resulting elements are all non-negative.38
We also use observable characteristics, namely ideology (measured by DWNominate from
VoteView), tenure (how many terms a politician has served in Congress, with data coming
from the Library of Congress), and committee memberships.
35Dividing the number of words by 100, reflects an appropriate scale to compare cosponsorships to thesespeeches. It is a reasonable scale as House rules explicitly limit one minute speeches, a useful tool forpoliticians (Schneider (2015)), to 300 words.36As there are changes in the composition of Congress within a term, for instance due to death or resignationamong other reasons, we have some observations whose cosponsorship numbers and word counts do notcorrespond to a full term. To mend this, we scale up values proportionally to the recorded behavior whilein Congress. In other words, if a politician leaves halfway through his term, we double the values of theseobservations.37Our qualitative results about the parameters still hold if we use either of these variables individually.However, the magnitudes of the estimates change due to the different scales of Roll Call Effort (between 0and 1) and the floor speech data (in hundreds of words).38Wang and Zhang (2013) provide a discussion of this methodology. NNMF works by factorizing a matrix, callit A, into two positive matrices W,H, under a quadratic loss function. The product WH is an approximationto A of smaller dimension, as there are less columns in W than rows in A. We then use the main factor inW as our proxy.
20 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Figure 2. Correlation between the raw data of log(1+Words) in FloorSpeeches and Cosponsorship decisions.
The figure shows the positive correlation between proxies for socializing (log number of cosponsor-ships/100) and legislative effort (log number of words in floor speeches/100). The graph presents thevariables in raw form, without rescaling or removal of members with low cosponsorship. The rawcorrelation is 0.171. In red, we present a LOWESS (locally weighted scatterplot smoothing) fit, withbandwidth (span) equal to 0.9, fitting the relationship between the variables. We do remove, as de-scribed in the Data section, observations that have total words equal to zero, which are mostly due todeath/resignations in that term.
Data on committee memberships comes from the work of Stewart and Woon (2016). To
quantify the value of the committees a politician is in, we use the Grosewart measure
(Groseclose and Stewart, 1998). Groseclose and Stewart (1998) and Stewart (2012) estimate
a cardinal value of how much an assignment to a given committee is valuable to politicians.
Such estimates are based upon data on how often politicians accept transfers from one
committee to another. The more desirable committees are those that politicians accept to
be transferred to often, but rarely accept to be transferred away from. The Grosewart
measure sums up the values of the committees in which a politician is present. We use the
estimates given in Stewart (2012) for our study, since they are the updated values for the
period we study.39
Summary statistics for all our variables of interest can be found for reference in Table 1.
39Below, we also consider an alternative measure for committee memberships. There, we construct dummyvariables for whether a politician has been assigned to a given committee during that congressional term.We then focus on the main committees for parsimony: Appropriations, Energy and Commerce, Oversightand Government Reform, Rules, Transportation and Infrastructure, and Ways and Means. We also includea variable Leadership of whether the politician was the Speaker, the Majority or Minority Leader, or theMajority or Minority Whip.
Number of Politicians N 442 435 440 439 438 445Number of Bills 4874 5681 5767 5431 6436 7340
The table presents summary statistics for the variables used in the structural estimation, across Con-gresses.40 Roll Call Effort is defined as the proportion of Roll Call votes that the politician does notappear as “Not Voting”. Number of words said in floor speeches aggregates the number of words saidby a politician across all his speeches in a term. Cosponsorships and number of words are scaled tofull term length (i.e. if a politician leaves mid-office and is replaced mid-office; then both him and thereplacement have those variables multiplied by 2.). For estimation, we remove the observations (billsand politicians) we do not have or cannot match to identifying numbers, and those with less than 3Cosponsorships (see the Data Section). These are mostly Congressmen who substitute others mid-term.Data used for bills is House bills (H.R.).
22 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
We restrict the data to Congresses 105th-110th for multiple reasons. First, the data
we employ to compute effort from floor speeches is only available from the 104th Congress
onwards. Second, the 104th Congress (corresponding to the Republican Revolution) provides
a structural break in the analysis of Congressional behavior. With multiple changes to
Congressional composition and structure during the 104th, it becomes hard to compare the
costs and socializing of this specific Congress to others, preceding or following, without
having to further delve into the exceptionality of this particular congressional cycle, which
is not the aim of this work.41
4. Estimation
4.1. Moment Equations. Let zi = [1, Ii∈P2 , z′i, z′iIi∈P2 ], where Ii∈P2 denotes a dummy
variable of whether politician i is in Party 2. Further define: βs = [log(S1), log(S2) −log(S1), β1, β2 − β1], βx = [log(X1), log(X2)− log(X1), β1, β2 − β1].
Appendix E shows that the moment conditions necessary to identify and estimate the
model’s parameters are:
Ezi(log(si)− z′iβs) = 0(4.1)
Ezi(log(xi)− z′iβx) = 0(4.2)
E(
2(log(si)− log(xi))− log(c− 1
X1
)−(log
(c− 1
X2
)− log
(c− 1
X1
))Ii∈P2
)= 0
(4.3)
(ElogP (yi = 1)− log(
1
ζ1
)− log(s2
i ))Ii∈P1 = 0.(4.4)
(ElogP (yi = 1)− log(
1
ζ2
)− log(s2
i ))Ii∈P2 = 0.(4.5)
S1 = φ1X1 (B1S1X1m11 +B2S2X2m12)(4.6)
S2 = φ2X2 (B2S2X2m22 +B1S1X1m12) .(4.7)
These moment conditions are based on rewriting the equations of Proposition 2.1 using
our parameterization for α and measurement errors, given in equations (2.14) and (2.16).
They allow us to identify (c, ζ1, ζ2, S1, S2, X1, X2, β1, β2) and set identify the φ parameters.
41In addition, without ad-hoc modifications to the estimating model specifically designed to accommodatethe idiosyncrasies of the 104th Congress, this lack of stability would also likely undermine any effort ofstructural estimation.We also perform a final, additional, trimming of the data across all Congresses. We drop a set of 19observations (out of 2636), that have the number of words in Floor Speeches set to 0 in the data of Gentzkowand Shapiro (2015). These observations relate almost exclusively to a politician who either resigned or diedduring that term (e.g. Representatives Jo Ann Davis in the 110th Congress, Sony Bono in the 105th, orresignations as Representative Bobby Jindal in the 110th). Since the data is zero, the rescaling above doesnot prove to be adequate, so we drop these observations. We also drop one observation in which politiciansthat have cosponsorship figures less than 3 bills over a full term, since identification relies on the existenceof cosponsorship and most cosponsor in the hundreds, so scaling is also inappropriate. The results do notdepend on this cutoff.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 23
Specifically, lack of point identification of (φ1, φ2) is the result of lack of point identification
of p1 and p2. The parameters p1 and p2 enter nonlinearly in equations (4.6) and (4.7) through
mij(s), identifying a ridge of (p1, p2) pairs satisfying (4.6) and (4.7).
In Appendix H, we demonstrate how to obtain point identification of the φs when we
impose additional restrictions on the proxy variables (si, xi). The restrictions there are on
the second moments of the effort proxies, similarly in spirit to random coefficient models.
Such restrictions are justified under the assumption that more partisanship may result in
noisier measurement of social interactions and legislative effort. This alternative approach
is heavier in terms of assumptions and for this reason we do not adopt it for the derivation
of our main results in Section 5. Instead, we report estimated values for all parameters
(including p1, p2) under these additional assumptions and specification in Appendix H.42
For our main empirical exercise, we let Party 1 denote the Democratic Party (with its vari-
ables denoted by the subscript Dem) and Party 2 denote the Republican Party (analogously
denoted with a Rep subscript).
We carry out the estimation process using a two step procedure. In the first step, we
compute estimates for the parameters (c, ζDem, ζRep, SDem, SRep, XDem, XRep, βDem, βRep) from
the moment equations (4.1) - (4.5) above, via GMM.
In the second step, we use the first-step estimates to derive a set estimate for (φDem, φRep).
This is done by using equations (4.6) and (4.7). We grid all pairs (pDem, pRep) ∈ [0, 1]× [0, 1],
and, employing the estimated ADem, ARep, SDem, SRep from the first step, we calculate the
values of mij for each pair (pDem, pRep). The set estimate for (φDem, φRep) are all the values
that satisfy equations (4.6) and (4.7) for any pair (pDem, pRep).
Concerning the information of whether a bill passed or not yi,τi∈N , the model is agnostic
on how many bills a politician proposes. Because a good fraction of members of Congress
sponsor multiple bills, however, we work with L > N bills in the actual data. This is easily
accommodated in the estimation. Recall that εb are i.i.d. across time and bills. For each
politician i, all i’s bills have the same associated network gi,j, as it comes from the same
politician and his same network and effort choices (as well as those of his network). The
different ε realizations, however, represent different bill qualities or institutional arrange-
ments within politician, meaning that the same politician may have one bill approved and
not another. The dimensionality of the problem can be decreased by simply averaging out
each bill’s success by politician. This is made possible by the fact that equation (4.4) holds
for all bills, implying that it must hold for all politicians as well. Hence, we use the average
pass rate of bills for politician i as its estimate of the probability of bill approval.
4.2. Estimation via GMM. To estimate the model, we replace equations (4.1)-(4.4) by
their empirical counterparts and stack them into a vector of the form 1n
∑ni=1 g(si, xi, yi, zi; θ).
Since all moments have expectations taken over εi, vi, which are i.i.d. and mean zero for all
politicians, the empirical counterpart replaces the expectation operator by the mean over
42As second moments restrictions allow us to obtain point estimates of p1 and p2, we use such estimates inthe last part of Section 5 for assessing additional implications of our model and validation.
24 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
i.43 Furthermore, we average over the approval rates for bills for each politician to get the
estimated probability of approval at the politician level.
We then minimize the quadratic form:
(4.12)
(1
n
n∑i=1
g(si, xi, yi, zi; θ)
)′W
(1
n
n∑i=1
g(si, xi, yi, zi; θ)
),
where 1n
∑ni=1 g(si, xi, yi, zi; θ) is given by stacking up the empirical counterparts of equations
(4.1)-(4.4), for a total of 2k + 2 equations (k being the dimensionality of zi). W is the
weighting matrix, which can be taken as the identity matrix (an inefficient choice), or the
optimal weighting matrix for the GMM.
Given these first stage estimates, we then estimate the set of feasible values for (φ1, φ2)
as previously described. Further details about the empirical implementation are discussed
in Appendix F.
5. Results
Table 2 presents our parameter estimates, which delineate a series of intuitive relationships
emerging from the data. Table 3 shows the distributions of the estimated direct benefits
from passing policy αi, over time and by party. These distributions appear stable across
Congresses.
Splitting the samples by party, we observe important differences in the estimated distribu-
tions of Republicans and Democrats. Democrats have a higher average and dispersion, while
Republicans have tighter distributions. This implies different social effort patterns across
parties, as Democrats socialize more and (by our social meeting function) more often with
other Democrats.
43That is, the expectation operator has one observation for each politician, and averages across all politicians.For example, the empirical counterparts to (4.1)-(4.2) are:
1
n
n∑i=1
zi(log(si)− z′iβs) = 0,(4.8)
1
n
n∑i=1
zi(log(xi)− z′iβx) = 0,(4.9)
or in matrix form:
Z ′(log(s)− Zβs)n
= 0,(4.10)
Z ′(log(x)− Zβx)
n= 0,(4.11)
where Z stacks up zi, and log(s), log(x) stack-up log(si), log(xi) respectively.
Notes: Standard errors in parentheses. The table presents the GMM estimates using the OptimalWeighting Matrix for the parameters of interest, as described in the Estimation section. Standard Errorsare computed from estimates of the variance for a GMM estimator with the Optimal Weighting Matrix.Details are in Appendix F. The estimate of φ1 and φ2 are their estimated sets. Rep represents the dummyvariable of whether a politician was in the Republican Party. Hence, a variable Tenure × Republicanrepresents the additional estimate of the Tenure variable for the Republican Party, as compared to theDemocratic one.
26 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Table 3. Differences in the Distributions of αi Across Parties
We show the mean and the standard deviation of the (estimated) distributions of αi for each party,highlighting the differences in those distributions. The estimates presented are those from Table 2.
From the main results, we see that both φDem and φRep have estimated sets that exclude
0. For Democrats, the estimates range from [0.037, 0.045], while for Republicans, the range
is [0.031, 0.036]. Both are slightly increasing over time. To put this into context, note that
the marginal utility of an increase in xi is
αi + φisi∑j 6=i
xjsjmij(s)− cxi.
The direct benefit αi ranges from 1 to 1.4, while the network benefit φisi∑
j 6=i xjsjmij(s)
ranges from about 0.1 to 0.3. So, the social incentive is somewhere between a tenth to a
fourth of the direct incentives, a substantial consideration in politician’s choices.
We note that the value for socializing, φDem, for Democrats is higher than that for Re-
publicans, even as we control for the difference in types, partisan bias and preferences across
both parties. This difference appears historical, not changing with majority status. It is
also quantitatively large, approximately 17% of the returns to social effort (in the order of
0.006/0.036 across Congresses). Furthermore, it is of a significant magnitude relative to c,
which ranges in [0.25, 0.28]) and does not change much over time, on average.
The relative cost of legislative effort to social effort, c, is stable over time. As a result,
interactions between politicians are more valuable, as there is an increasing return of social-
izing against a stable cost. This may be consistent with an increase in the complexity of
extant statutes, making it more difficult to approve legislation. This is for example evident
from an average number of pages per statute of 3.6 in 1965-66 to 18.8 in 2015-1644, making
interactions between politicians in drafting and drumming up support for legislations on the
Capitol more important. Although this would not change the costs to social effort, it would
appear to change the returns from it.
The estimates of ζDem and ζRep are also significant and large in magnitude. This indicates
that politicians see positive gains from having bills approved. The larger magnitude of ζ
44Vital Statistics of Congress 2017, Chapter 6, available at www.brookings.edu
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 27
relative to the other parameters is needed for the model to be internally consistent. This is
because ζ also operates as a normalizer that guarantees that the probability of bill approval
(in equation (4.4)) is between 0 and 1. However, we can see large differences across parties.
In particular, the Democrats have the largest ζ between Congresses 105-109, when they are
in the minority. In Congress 110, this is reversed, with the minority Republicans having
the largest ζ. This intuitively suggests that the returns from getting bills approved for a
politician depend on their party’s majority status in the House. A high ζ for the minority
party indicates that voters reward legislators more when it is harder for them to pass bills.
This is consistent, for instance, with voters learning from legislative successes about the
quality of their representatives.
Using the estimated values of ζP (i) for both parties, we can also calculate the probability
of bill approvals for each politician. We show these in Figures 3a for Democrats and 3b for
Republicans.
By comparing Figures 3a - 3b with the average bill passage rates in the summary statistics
(Table 1), we can see that the model can generate a good match of the mean bill success rates
(which we observe). Our structural assumptions allow us to represent the whole distribution
of expected probabilities of having a bill approved across different politicians. These indicate
some variation over time. Later Congresses (108th and 110th) show a higher predicted
approval rate for most politicians. Furthermore, approval rates are highly correlated with
majority party status. Democrats have a much higher rate as a majority in Congress 110,
while Republicans have higher ones as a majority in 105-109.
We can also discuss the statistical significance of different covariates in explaining direct
benefits, αi. With our baseline specification that uses Ideology, Tenure, Grosewart for zi in
Table 2, we see that ideology is statistically significant (especially in later Congresses). The
estimates suggest that those on the left of the ideological spectrum (extremist Democrats,
moderate Republicans), have higher direct returns of exerting legislative effort. Meanwhile,
the Grosewart variable, capturing the impacts of committee assignments, appears to be
noisy.
We also consider another specification where we replace the Grosewart variable by dummy
variables of committee assignments to each of Congress main committees. This is shown in
Table C.3 in Appendix. We can see that the results from our main specification are robust.45
Finally, we investigate the relationship between estimated party types (AP (i) =∑
i∈P (i) αi)
and partisan bias in social interactions, pi.46 Figure 4 reports the results for Democrats and
Republicans separately.
45It is noteworthy that in this specification, the estimate of being in the Rules Committee is positive andsignificant. The Rules Committee is the committee in charge of determining the rules that allow each billto come to the floor, fundamental for the progress of legislation. It seems consistent that politicians in thatcommittee are rewarded for effort in it, even conditional on having the same ideology, party, and tenure.46Absent an estimate for p1 and p2 from the baseline model where these parameters are not identified, weemploy the estimates generated by a specification imposing second moment restrictions, reported in AppendixH. As discussed in the next section and in Appendix, these estimates appear to fit empirical patterns in thedata better than corner solutions.
28 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Figure 3. Estimated Probability of Approval
(a) Democrats
(b) Republicans
We find a clear negative correlation between types and partisan bias for Republicans, even
with our small sample of Congresses, while the relationship is absent for Democrats. Our
theory does not provide a microfoundation of the partisan bias of each party, yet an intuitive
explanation for these two relationships may arise from the model’s internal mechanism.
Recall that social externalities are stronger for the higher types, who have larger direct
benefits from legislative activity and a larger social network to benefit from. Republicans
benefit from socializing with Democrats who are higher types on average. As Republican
types increase, then they will choose higher social effort and may benefit even more from
bipartisan connections with the Democrats. This would encourage a lower partisan bias for
GOP as the average type of its politicians increases. At the same time, the same incentives
may not necessarily hold for Democrats and this mechanism can reverse. Democrats have
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 29
Figure 4. Relationship between Partisan Bias and Estimated Types
(a) Democrats (b) Republicans
The figure shows the relationship between estimated party types (the sum of each party’s estimatedα’s from Table 2, defined as A1, A2 in the model) and the partisan biases (estimated as described inAppendix H).
the highest types on average, so the largest externalities are when their legislators interact
only within their own party. With more segregation across party lines (higher partisanship
pi), Democrats may deal with higher types relative to when also mixing with Republicans.
This suggests a possible opposing incentive in selecting a low pi for Democrats as their direct
incentives grow and it can rationalize the mixed evidence in the Democrats’ panel of Figure
4. The reader can find additional discussion of the problem of endogenous partisanship in
Appendix.
5.1. Fit and Further Discussion. To conclude the section, we conduct three exercises. In
the first, we consider an out-of-sample validation of our approach, by assessing the fit of our
model of moments of the socialization patterns not used in estimation. In the second, we
look at an in-sample validation comparing the fit of empirical approval rates of bills to the
estimated values. In the third, we compare the prediction of legislative behavior according
to our model generated network to other salient examples of political networks used in the
literature, including cosponsorship networks, alumni-connections networks, and those based
on committee memberships.
Although our analysis employs i’s total cosponsorships to proxy for his/her social effort
s∗i,τ , the more fine-grained data on pairwise cosponsorship information between i, j politician
pairs is not used in estimation. In this first exercise, we predict the i, j links for each
pair of politicians based on what predicted by our theoretical gi,j(s) function based on the
estimated parameters for each Congress in our sample and then compare them to the actual
cosponsorship pairs. The goal is to show that our estimated network model fits such proxies
for social ties, commonly employed in the literature following Fowler (2006), even without
The correlations between the estimated gi,j(s) and any i, j pairwise cosponsorships are
reported in Table 4. The Table illustrates correlations for two possible definitions of links
based on actual i, j cosponsorship in the data. In the top panel cosponsorships are consid-
ered directed from i to j and in the second panel cosponsorships are considered a-directional.
In the two cases the correlations with the model-implied gi,j(s) are 0.375 and 0.456 respec-
tively and statistically significant. Thus, the model appears able to capture disaggregated
socializing proxies not directly targeted in estimation, reassuring on the plausibility of our
socialization structure.
We can draw an additional relevant conclusion from this exercise. Results of Fisher’s
z-transformation tests also suggest that our model with pDem > 0, pRep > 0 is better at
capturing the relationships from the pairwise cosponsorship data than alternative models
with either full partisanship (at least one of pDem = 1 or pRep = 1) or without partisanship
(pDem = pRep = 0). These comparisons are possible as different gi,j(s) can be generated
using different values for pDem, pRep.47
Although recent political economy research highlights a hollowing out of the moderate
middle ground in congressional voting (Fiorina, 2017; McCarty et al., 2006; Canen et al.,
2020), our model with pDem, pRep around 0.1 (values in this range for pDem, pRep are obtained
in Appendix H) produces a substantially better fit of the cosponsorship data than a model
with complete polarization pDem = pRep = 1, which is statistically dominated. Also, while
the exact point estimates of pDem, pRep rely on our assumptions on second moments, we
believe that the rejection of pDem = pRep = 1 has to be considered more general. The raw
data in Figure 1b display a sufficient degree of cross-party cosponsorship to cast doubt on an
hypothesis of “full sorting” among party members in the House. A model with full polariza-
tion would predict 0 relationships emerging across different parties, that is obviously not the
case. Meanwhile, a model with no polarization would predict a much higher degree of social
connection across party lines than observed (it predicts approximately 60% of Democrat
connections to other Democrats, compared to around 70% for our model and the data; and
fewer connections among Republicans alone, relative to Republicans and Democrats, when
compared to our model and the data).
Possibly, reconciling a world of more polarized legislators and the thousands of cospon-
sorships across party lines reported in Figure 1b, may come from noting that, as ideology
may diverge, engagement across party lines becomes more important for getting legislation
to the floor and passed. Our model appears to capture such phenomena.
In a second exercise, we fit the empirical success rate yi for each politician in each Congress
to the estimated one, given by equations (4.4)-(4.5).
For this exercise, we focus on politicians who have sponsored a sufficiently large amount
of bills (over 10 House Bills sponsored), so to have a reasonable empirical approximation to
their average success rate. The results are shown in Figure 5. We find a significant positive
correlation of 0.265 between estimated and empirical bill success rates. Thus our model
47As before, we use the estimates of p1 and p2 generated by a specification imposing second moment restric-tions, reported in Appendix H. This is because these parameters are not identified in the baseline model.Nevertheless, this test shows that our model captures part of the empirically observed relationship.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 31
Table 4. Model Fit: Correlation of Estimated Network of the Model to theCosponsorship Networks in the Data
Congress Correlation Fisher’s z-statistic
Data from Directed Cosponsorships:
Model: pDem > 0, pRep > 0 0.375 -
Model: pDem = 0, pRep = 0 0.365 9.427***
Model: pDem = 1 0.349 22.725***
Data from “Combined” Cosponsorships:
Model: pDem > 0, pRep > 0 0.456 -
Model: pDem = 0, pRep = 0 0.443 12.713***
Model: pDem = 1 0.426 27.891***
We compare the performance of the partisan model (with pDem > 0, pRep > 0) to the performance of themodel without partisanship (pDem = pRep = 0) and complete partisanship (pDem = 1), in explainingthe observed cosponsorships in the data. In the first panel, cosponsorships are measured by the directednumber: how many times i cosponsors j. In the second panel, “combined cosponsorships” are measuredby the number of times i cosponsors j and j cosponsors i, creating a symmetric undirected graph.To calculate the statistics for each model, we first generate the links using the theoretical definitiongij(s) = sisjmij(s) under our estimated parameters. That is done for the 3 cases. We then show thecorrelations of the model links to the values of the cosponsorships in the data. The estimated valuesfor pDem > 0, pRep > 0 come from the estimates using second moments of (si, xi), from Table H.1in Appendix H. We present Fisher’s z-transformation statistic, for the test that the correlation of theadjancency matrix of the Model with pDem > 0, pRep > 0 with the data is equal to the correlation ofthe alternative model (without partisanship/complete partisanship) with the data. Since our modelgenerates a symmetric adjacency matrix by construction, we consider the correlations of the lowertriangular adjacency matrices. ∗∗∗ represents that the null hypothesis of equal correlations can berejected at 1% significance level, ∗∗ at 5%. Note that, when estimating the model, we did not usecosponsorships at the ij level. We aggregate all Congresses in the analysis above.
captures part of the empirically observed relationship. Reassuringly, as we use more precise
measures of the empirical success rate of politicians, this correlation tends to increase.48
Finally, we conduct a horse race comparing the in-sample properties of our endogenously
estimated network to exogenous alternative G’s in the literature. We compare these networks
in terms of their predictive power/model fit of legislative behavior using the outcome from
48The correlation is stronger as we increase the threshold of 10 sponsorships. It is 0.283 if we look atpoliticians with over 20 sponsorships, 0.362 for those over 30, and 0.373 for those over 50.
32 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Figure 5. Model Fit: Estimated and Empirical Approval Rate of Bills
The figure shows the correlation between estimated and empirical approval rates for politicians. Weconsider politicians with over 10 sponsorships, so that a meaningful estimate of the average approvalrate can be obtained. The correlation is 0.265 in this case, and is stronger when we look at politicianswith a higher number of sponsorships.
our main specification. To do so, assume that gij(s) is known, as we will feed it from the
data. Then, the game defined by (2.1) collapses to that in Ballester et al. (2006), with the
Nash equilibrium given by:
(5.1) (I − φG)x∗ = α,
where α = (αi)i. Using our measurement error assumption of x∗ in (2.14) and the
parametrization of α in (2.16), equation (5.1) can be rewritten as:
log(xi) = log((I − φG)−1ez′iβ) + vi.
≈ log((I + φG+ φ2G2 + φ3G3)ez′iβ) + vi,(5.2)
for small φ and G of (approximate) full rank. We can then compare the fit of equation
(5.2) across different networks G, as this model does not include any of the endogenous
network structure from our own model. We compare the predicted endogenous network from
our model to graphs based on directed cosponsorships (Fowler, 2006), alumni connections
(Battaglini and Patacchini, 2018; Battaglini et al., 2018), and committee membership (from
the main committees used in our estimation).49
49For illustration purposes, we present figures of these networks in Congress 110 in Figure C.1 in Appendix.From those graphs, one can check that the alumni and committee-based networks are much sparser thancosponsorship-based networks. For example, approximately 270 legislators are isolated in the alumni network,
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 33
To estimate φ, β in (5.2) across models, we first look at the subgraph of non-isolated
nodes for each case. We estimate those parameters by Non-Linear Least Squares using
the covariates of Ideology, Tenure and Grosewart, used in our main specification of Table 2.
Then, we use the estimated parameters to fit the Mean Squared Error across all observations
for each alternative network G. Note that covariates are essential in this specification so that
one can generate heterogeneity in αi.50 The results for this exercise are shown in Figure 6.
Figure 6. Comparison of Model Fit Across Our Model Network and Competitors
The figure compares the performance in terms of mean squared error (MSE) of different legislativenetworks used in the literature (our estimated model, one based only on directed cosponsorships, onebased on university alumni networks, and one based on committee memberships) in fitting legislativebehavior (our xi). To do so, we estimate the Nash equilibrium of the game in which the network G isgiven, which collapses to the problem in Ballester et al. (2006).
Our endogenous network outperforms all competitors in explaining legislative behavior in
terms of MSE. For example, our model fits legislative behavior much better than the sparser
alumni and committee networks. This is because it can capture correlated behavior across the
hundreds of nodes that those alternative models assume are isolated, but that in reality are
highly correlated, better captured by our endogenously formed network. While alumni and
with an average degree below 2, while no legislator is isolated in the cosponsorship case. The data for thecommittee and cosponsorship based networks is the same as used in estimation in the previous section.The committee network has edges defined by whether legislators sat on the same committees. For alumninetworks, we scrape the Congressional bioguide webpage and use fuzzy matching based on the politician’suniversity and date of graduation to generate the network. A link on this network exists if politicians wentto the same university within 8 years of one another (as in Battaglini and Patacchini, 2018).50Monte Carlo simulations have found that φ, β can be recovered reliably in this set-up. Nevertheless, thesesimulations also illustrate the empirical limits of a model based on (5.2). For example, discrete covariatesgenerate identification problems due to the lack of variation in the support of those variables. For this reason,we only keep the covariates from our main specification in this exercise.
34 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
committee networks can fit the behavior of very central and influential nodes, they miss out
on the majority of legislators who do not have such easily codified connections. In addition,
our model appears to outperform even a pure cosponsorship network. The cosponsorship
network can appear too dense, since observed cosponsorships might be only weakly related
to underlying relationships. Meanwhile, our model uses cosponsorship information more
efficiently to find underlying links, as it is able to explore how cosponsorships correlate
with other legislative behavior through equilibrium restrictions. We note that this is a fair
comparison across models - while we do use cosponsorships to inform our model, we only
use aggregate information about one’s cosponsorships. However, the directed cosponsorship
network in the alternative uses information about pairwise decisions to completely determine
G.
Altogether, this last exercise further illustrates the benefits of simultaneously modeling
simultaneous network formation and strategic interactions on that network.
6. Multiplicity of Equilibria and Counterfactuals
Our theory allows for multiplicity of equilibria. In this section, we first discuss what we
can say about the equilibrium being played in each Congress. In the second part of this
section, we present counterfactual exercises using the estimated model parameters.
6.1. Stability of Equilibrium and Other Equilibrium Properties. Let us first discuss
the stability of the equilibria that we find.
A preliminary technical consideration deals with the fact that we only infer noisy esti-
mates of social and legislative effort (SDem, SRep, XDem, XRep) from the data. Those values,
however, do not necessarily solve the original (exact) system defined in Proposition 2.1 under
our set of estimated parameters, they only approximate its solution. To find the equilib-
rium values (SDem, SRep, XDem, XRep) that are consistent with our estimated parameters for
(c, φDem, φRep, βDem, βRep), we solve the system in Proposition 2.1 exactly.51 Those solutions
are presented in the upper panel of Table C.4 in Appendix. They are used to compute the
model-consistent bill approval rates shown in Figures 3a - 3b, as well as the counterfactuals
presented below. Based on the results in Figures 3a - 3b and in the upper panel of Table
C.4, Congress is always at an interior equilibrium of our model.
As there can be multiple interior equilibria, we check that the estimated equilibria are
stable.52
51This procedure is described in more detail in Appendix G. For example, for φs, we use the median valuein their set.52Generally, when there are multiple interior equilibria, only some are stable. This is in contradiction withProposition 1 in Cabrales et al. (2011) which claims stability of all interior equilibria. In their model, contraryto the original proof, the largest equilibrium is unstable. In the proof of that proposition the matrix Π cannotbe approximated by setting off-diagonal terms to 0. In fact, the eigenvalue can change sign if the off-diagonalterms are included and are on the order of 1/n. This reverses their conclusion.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 35
To perform this analysis, we use the best response dynamics.53 Starting at some vectors
s0, x0 and iterating through t, one obtains that the best response dynamics are described by:
sti = xt−1i φi
∑j 6=i
mij(st−1)st−1
j xt−1j ,
and
xti =αic
+ st−1i
φic
∑j 6=i
mij(st−1)st−1
j xt−1j .
Then we check whether perturbations away from equilibrium socialization and legislative
efforts converge back to the estimated equilibrium efforts through the best responses of the
players in the network. This is done by starting slightly below or above the values of the
low equilibrium and successively iterating the best response dynamics.54
In all Congresses 105th-110th, we find that best-response dynamics converge back to our
estimated equilibrium (from the upper panel in Table C.4) after few iterations.55
Second, we compare our estimated equilibrium effort levels to those from a possible equi-
librium with higher levels of effort in each Congress. To do so, we take the system defined
by Proposition 2.1 under the estimated parameters and numerically search for solutions of
this system at higher values of (SDem, SRep, XDem, XRep).
These results are reported in the lower panel of Table C.4 in Appendix. We find that
there exists a higher effort level equilibrium for every Congress considered in our analysis.
These other equilibria are distinct from the ones we estimate, with effort levels that are
approximately 20% larger in magnitude to the empirically assessed ones. We also verify that
these unobserved equilibria are unstable. To do so, we repeat the stability exercise, starting
below the values of the higher equilibria. Here, we find that the dynamics diverge away from
these higher equilibria. Given that this is the only equilibrium with higher effort levels in
our set-up (see footnote 23), we don’t have a stable equilibrium with higher effort levels with
our parameters.
From these exercises one deduces that Congress is generally in an interior, low-effort
equilibrium and, moreover, that all Congresses operate at effort levels lower than at an
unobserved, unstable, high-effort equilibrium. These equilibria Pareto dominate the observed
equilibria.56
53For details, we refer the reader to Appendix B. Here sti takes xt−1i as given, but one could also solve for
simultaneous best reply dynamics and get the same results.54This is not a full check of stability, as we are not verifying all perturbations. However, given the structureof equilibria, all best responses in a party are proportional to the same X,S and have similar dynamics.55Only an adjustment is needed for Congress 107, where the pointwise estimate for c generates a numericalinstability. Other values within its confidence interval confirm this result.56We conclude this discussion by briefly noting that there also always exists a “semi-corner” equilibrium.In this equilibrium legislative effort is exerted and is chosen at level x∗i = αi/c, but there is no socializing,i.e. s∗i = 0 for all i. This occurs since there is no return to social effort if no other politician is socializing.Each politician acts in autarky. Effort is still provided in the model, because there are direct incentives αifor legislative effort, but no law is passed. Such an outcome is not desirable for politicians or voters due toProposition 2.3. Furthermore, this semi-corner equilibrium is unstable, in the sense that, were any politicianto deviate to a positive social effort sj , so would all the other politicians. The semi-corner equilibrium is not
36 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
6.2. Counterfactuals. We use the model to assess four counterfactuals.
First, we investigate the topical issue of how social networks vary endogenously with
political polarization and its consequences for legislative activity. We consider an exercise
in which there is an increase in polarization stemming from either drifts in the ideology of
certain members of Congress or from partisanship in mixing with politicians from a different
party. We then discuss how those results can be overturned if the selection of politicians to
the House of Representatives changes.
Second, we continue with a counterfactual analysis of the Congressional emergency re-
sponse to the 2008-09 financial crisis by modifying the partisan composition of the 110th
House back to the 109th GOP majority. This counterfactual provides evidence against a
common claim during the Tea Party movement surge of 2010 that excessive government in-
tervention during the crisis was rooted in the Democratic party’s congressional control. We
show that a counterfactual GOP controlled House would have not had much lower likelihood
of passing the same legislation in the fall of 2008. The final two counterfactuals investigate
the nonlinear properties of the model. We briefly describe them at the end of the section
and present the full results in Appendix.
6.2.1. The Effects of Polarization on Effort Levels. Do social and legislative activity in Con-
gress decrease with political polarization? What are the changes in the likelihood of bill
passage and legislative activity when bipartisanship becomes more rare? And, if the effects
of polarization are welfare-reducing, can they be overturned? Our model provides guidance
in answering these questions.
In our setting, polarization can be interpreted in two distinct ways. First, polarization
can be the drifting apart of ideologies of individual legislators. Second, polarization can map
into partisanship: a reduction in the likelihood that members of two parties mix socially as
opposed to only within their own party.
We begin with the first approach in changing the relative ideologies. In our empirical
model, type αi correlates significantly with ideology and in fact the direct benefit from
legislative effort appears to decrease as ideologies drifts to the right (Table 2). This is the case
when Republicans become more extreme, for example, as it has been extensively documented
in a large literature on ideological polarization in the modern Congress (see McCarty et al.,
2006). Our first exercise looks at the effect on equilibrium activity level of Republicans’
ideological stances shifting to the right by 20% holding the ideology of Democrats constant.
This translates in lower direct benefits from legislative activity for Republicans by 10%.57
As we see from Table 5, an increase in polarization leads to an average decrease of 8.3%
and 7.5% in social effort, for Democrats and Republicans respectively, and of 3.4% and 1.9%
for legislative effort. Intuitively, Republicans now have lower benefits, so they provide less
observed, given that we observe positive socialization in Congress. Such a complete shutdown of socializationeffort would be unstable, and so should not be observed for any length of time.57An increase of approximately 20% in ideological stance of for Republicans correlates with a decrease of10% in their direct benefit from legislative activity. To see this, recall that αi = ez
′iβ , the magnitude of
zi = Ideology is of the order of 1 and we seek the coefficient ∆ such that 0.9 = e∆βIdeology . For a value ofβIdeology at approximately -0.5 (Table 2), we get that ∆ = −ln(0.9)/0.5 ≈ 20%.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 37
Table 5. Effects of Polarization on Equilibrium Effort Levels (and How toOverturn it)
Congress 105 106 107 108 109 110
Baseline Equilibrium
SDem 0.922 1.506 1.443 1.403 1.412 1.484
SRep 0.724 0.916 0.808 0.831 0.881 0.988
XDem 4.478 5.680 5.111 5.058 5.206 5.484
XRep 4.231 4.773 4.196 4.235 4.426 4.722
Change in Effort Levelswhen ARep decreases by 10%
SDem -0.070 -0.130 -0.095 -0.109 -0.123 -0.152
SRep -0.056 -0.071 -0.047 -0.056 -0.067 -0.093
XDem -0.092 -0.214 -0.150 -0.170 -0.194 -0.246
XRep -0.063 -0.094 -0.057 -0.069 -0.086 -0.127
Change in Effort Levelswhen ARep decreases by 10%,but ADem is 10% higher than estimated
SDem 0.053 0.223 0.156 0.231 0.269 0.233
SRep 0.023 0.156 0.135 0.157 0.173 0.165
XDem 0.071 0.381 0.251 0.372 0.441 0.392
XRep 0.027 0.220 0.174 0.207 0.237 0.239
We compute the effects of polarization on effort levels in each Congress. In the second panel, we decreasethe types of Republicans by 10%, representing a drift to the right of their ideologies by about 20%. Weshow the changes to (SDem, SRep, XDem, XRep) relative to the values in the baseline (first panel). In thethird panel, we show the changes to equilibrium effort levels when Republican party types decrease by10%, but when the types of the Democrats (ADem =
∑i∈Dem αi) increases by 10% relative to estimated
levels. Although polarization decreases effort levels relative to the estimated values, it is sufficient forone party’s types to increase moderately to generate spillovers sufficient to increase effort levels. Thisis even the case when Democrats are the minority. Further results are in Appendix (Table C.5)
legislative and social effort in equilibrium. This also impacts Democrats, however, whose
socialization with Republicans involves dealing with even lower types, lower effort levels and
less dense social networks. There are fewer externalities from social connections, decreasing
the returns from social activity for all politicians. Since legislative effort is complementary,
38 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
it decreases for all legislators, but not equally. The impacts on legislative productivity are
even greater for Democrats, who are more active legislators and higher αi to begin with.
What can be done to reverse the perverse effects of polarization, described in Table 5? It
turns out that even small increases in the types of elected politicians within one party can
generate enough social activity to neutralize increased party bias. In Table 5, we show that
an increase in 10% in the average type of the politicians in the Democratic party is sufficient
to generate equilibrium effort levels above the baseline of no polarization (i.e. before the
initial change in GOP occurred).58 Note that this is true even though Democrats are the
minority party in most of the sample. This is driven by spillovers across parties from having
high αi types in Congress. If Democrats’ types increase, they interact with each other
and with Republicans more (under pDem < 1). This implies that there are higher returns
for Republicans to provide more social effort to connect with those active Democrats. All
legislators, in turn, increase their legislative effort. The equilibrium effects of this increase
in types are amplified in a large enough way to overturn the initial decrease in effort due to
polarization.
We note that social connections are the key mechanism by which the behavior of more
active members can permeate throughout Congress, leading to gains from one party reaching
all legislators. Other models that treat legislators as independent actors would miss such
externalities from strategic behavior.
A policy takeaway from this exercise is that unilateral improvements in the screening of
candidates by one party could neutralize the effects of polarization in the aggregate. These
improvements do not need to be coordinated, as decisions by one party can be sufficient. In
practice, such policies may include the adoption of different recruitment procedures for new
candidates, different primary election rules, or changes in party campaign financing. Finally,
even if some of these policies were to increase polarization (see Barber and McCarty, 2015 for
a review on the evidence of primaries driving polarization, for example), our results suggest
that the externalities of these policies might be large enough for a resulting positive effect
on legislative productivity.
A second interpretation of polarization within our model comes from changing partisan-
ship, via the pis. Polarization then corresponds to the hollowing out of bipartisanship,
captured by increases in (pDem, pRep). This is another form of polarization, as Democrats
and Republicans meet less frequently and connect less with one another to approve legisla-
tion. The incentives under this type of polarization differ from the case of ideological drift
and produces a somewhat more ambiguous interpretation of the effects of polarization on
legislative effort and productivity.
Numerical exercises are shown in Table C.5 in Appendix, where we increase pRep to 0.3
relative to baseline estimates of 0.05 − 0.15.59 The results suggest that polarization itself
is not necessarily effort reducing. In fact, polarization may lead to higher effort levels for
58We note that this argument holds for values below 10% as well, given that the gains of such a change arelarge enough (see the third panel in Table 5). We stick to 10% as a focal number.59Changing both parties’ biases to 0.3 instead of just the Republicans’ does not significantly change theresults.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 39
one party than in the non-polarized equilibrium. As a result, that party also obtains higher
approval rates for legislation.
The intuition relies on having one party with types that are sufficiently higher than the
other (ADem >> ARep), in line with what is estimated in the data. This type of polariza-
tion makes legislators from the high type party interact mostly within their own party.60
This isolation generates large externalities from connections, as high type politicians con-
nect more often with other high types within their party. This outcome may be preferable
to dealing with members of the opposition (low types on average), who choose low effort
levels, do not have as many connections and, as a consequence, do not generate many
spillovers/externalities from interactions.
Table C.5 suggests that in this case reducing bipartisanship may be welfare enhancing, as
long as one party has sufficiently higher αi than the other, and it would benefit to work only
within their group to get legislation approved.61
6.2.2. Counterfactual of the Democratic Party Takeover in the 110th Congress and Tea Party
narrative. We now propose a counterfactual of congressional behavior during the 110th cycle.
Elected in November 2006, the House of Representative turned Democratic majority after
twelve years of consecutive Republican control. This revealed to be a particularly consequen-
tial election, as it was the 110th Congress that voted between the Summer and the Fall of
2008 a host of emergency economic measures in response to the 2008-09 financial crisis (for
an analysis of these Congressional votes, see Mian et al., 2010). Some of this legislative ac-
tivity happened to be extremely momentous, including the vote of the Emergency Economic
Stabilization Act of 2008 (EESA, also known as the “TARP” from the Troubled Asset Relief
Program), which initially failed passage in the House, inducing one of the largest intra-day
losses in NYSE’s history.
An important counterfactual is to assess how relevant the role of congressional networks
was in eventually guaranteeing a responsive legislative intervention to the financial crisis.
How different would legislative activity have looked absent the Democratic party takeover
in 2006?
Within our framework, this counterfactual corresponds to keeping the composition of the
109th Congress in the 110th Congress. That is, we keep the observed characteristics and
estimated αi for the members of the 109th Congress in an analysis of outcomes in the 110th
Congress. Meanwhile, the institutional setting of the 110th Congress remains the same (with
the cost and returns to social effort, as well as the other institutional parameters kept at
their estimated values for the 110th Congress).
Table 3 implies that the distribution of αi from the 109th Congress appears mostly mildly
to the left of what is estimated for the 110th Congress. However, even small differences in the
vector of α could potentially produce substantial effects through the network in terms of bill
likelihood of success. In Table 6 we inspect the magnitude of the counterfactual reductions in
the likelihood of bill passage for some of the most important emergency response legislation
60A particular example is the case of full polarization, also shown in Table C.5 in Appendix.61This suggests that parties would have incentives to affect bipartisanship. At the end of Appendix B wediscuss endogenous partisanship in this model.
40 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
during the Fall of 2008. This set includes in addition to the EESA, the Housing and Economic
Recovery Act of 2008 (aimed at foreclosure prevention, also studied by Mian et al., 2010),
the Economic Stimulus Act of 2008 and the Supplementary Appropriations Act, both large
bills precursor of the fiscal intervention of 2009. Table 6 reports the relative differences in
bill passage probabilities between the counterfactual and the estimated model, as well as the
baseline probabilities.62
It appears all differences are in the range of a 10−15 percent reduction in the likelihood of
success, a quantitatively small effect considering the baseline probability of approval. That
is, the social network composition of the House would have not changed in a sufficiently
different way to substantially affect final voting outcomes. This is counter to the claim that,
absent the Democratic takeover of the House in 2006, the financial crisis response would
have been substantially different, with a more restrained government intervention under
a Republican Congress, taking as given emergency legislation being set forth by Treasury
Secretary Hank Paulson and Federal Reserve Chairman Ben Bernanke. This is a relevant
consideration, as it speaks to one of the crucial claims behind the Tea Party movement surge
within the Republican party in 2010, whose primary targets were government intervention
during the crisis and the consequences of the emergency stabilization and stimulus programs
on U.S. national debt (Mayer, 2016).
In Appendix C, we provide two additional counterfactuals that assess basic properties of
the model, first by changing the relative cost of legislative effort c, and then by reducing
bill quality (but not the incentives to socialize - that is, changing γP (i) while keeping φiconstant). These two exercises further emphasize nonlinear effects at play within our model
- in the first, increases in c lead to large nonlinear decreases in the probability of bill approval.
The mechanism is through the complementarity between legislative and social effort: as the
first becomes more costly, it is reduced, which by complementarity induces decreases in the
equilibrium levels of the other form of effort. Through feedback, as all politicians incorporate
this, the new equilibrium has lower effort levels in both dimensions. Meanwhile, the second
exercise finds that simple changes to average bill quality without changing socialization
outcomes do not generate externalities in effort. From that, we conclude that reforms aimed
at increasing effort levels and bill approval rates should alter the incentives to socialize to
generate nonlinear externalities - through the distribution of types, the relative costs of
socialization c, or the relative benefits from it, φi.
62Some of these bills’ complex histories align with a low likelihood of success. The EESA of 2008 failedthe House. In addition, about H.R. 3221, Mian et al. (2010) write: “Roll call 301: “On Agreeing to theSenate Amendment with Amendment No. 1: H.R. 3221 Foreclosure Prevention Act of 2008. This vote isconsidered by many the first crucial roll call in the political economy of the crisis and was characterized bystrong opposition (and a veto threat) by the executive branch. The Wall Street Journal (May 9, 2008) refersto the vote as follows: “The House voted 266-154 in favor of the centerpiece of the legislation $300 billionin federal loan guarantees -despite a White House veto threat. In particular, “The heart of the legislationis a program to help struggling homeowners by providing them with new mortgages backed by the FederalHousing Administration. The guarantees would be provided if lenders agree to reduce the principal of aborrower’s existing mortgage. H.R. 3221 had also previously failed cloture in the Senate in February 2008.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 41
Table 6. Counterfactuals in α: Looking at the Changes in (Ex-Ante) pre-dicted probability of Emergency Crisis bills in the 110th Congress, if the Re-publicans who lost their seats remained
Act Proportional Baseline ProbabilityChange of Success
Emergency Economic Stabilization Act of 2008. (H.R. 1424) -0.119 0.165Sponsor: Patrick Kennedy, Democrat - RIHousing and Economic Recovery Act of 2008 (H.R. 3221) -0.110 0.175Sponsor: Nancy Pelosi, Democrat - CAEconomic Stimulus Act of 2008 (H.R. 5140) -0.110 0.175Sponsor: Nancy Pelosi, Democrat - CASupplementary Appropriations Act, 2008 (H.R. 2642) -0.120 0.141Sponsor: Chet Edwards, Democrat - TX
The table presents the proportional change (Counterfactual - Model)/Model of the probability of eachbill passing under our counterfactual scenario. The counterfactual scenario is keeping the Republicanmajority and composition from the 109th Congress, in the 110th Congress. To do so, we keep thecharacteristics of all politicians from the 109th Congress with the estimated parameters of the 110thCongress. The only difference to characteristics is we add 1 for each politician’s Tenure variable, asthose politicians would have stayed 1 extra term in the counterfactual. We do not change the Committeecomposition or ideology. We then calculate the projected probability of bill approval using the estimatedparameters from the 110th Congress. The baseline (model) probability is shown in the second column,computed using pDem, pRep from Table H.1 in Appendix H.
7. Conclusions
We have developed and estimated a structural model of legislative activity for the U.S.
Congress in which endogenous, partisan social interactions play an important role in pro-
moting bill passage. We estimate that social effort matters substantially and significantly
for legislative activity.
By endogenizing both legislative and social efforts, we are able to accommodate comple-
mentarities in actions that appear to be strong. In particular, we find that complementarities
among politicians are quantitatively substantial (on the order of 0.1 to 0.25 of the direct in-
centives), and are fairly stable across our sample period.
We also find that the two parties have different base payoffs from passing bills, both in
terms of the average and variance across party members (both are higher for the Democrats).
Overall, we show how the process of informal social interaction among legislators may paint
a less extreme, although still partisan, picture of the internal operation of Congress.
Multiple equilibria arise naturally within our theoretical setting (as it is typical of models
of endogenous network formation). By careful consideration of the theoretical model and its
behavior around the estimated equilibrium, we are able to show that Congress appears in a
stable, low-socialization equilibrium, with effort levels lower than in a Pareto superior, but
unstable, equilibrium present in all Congresses.
Finally, our estimated model enables us to perform relevant counterfactuals. We show
how polarization can lead to decreased social and legislative activity in Congress, but that
can be overturned by a unilateral improvement in the selection of politicians. Politicians
42 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
who are more active can generate large enough externalities from their behavior that spills
over across parties. We also estimate that the response to the 2008-09 financial crisis would
not have changed in terms of levels of legislation if the Democrats had not taken over the
House and show substantial impacts from changing the relative cost of social effort in terms
of probability of bill passage.
From the methodological perspective, our tractable model of endogenous network for-
mation with biased socialization may be fruitful for further investigation of the behavior of
other political agencies beyond the U.S. context or in more general environments where both
endogenous socialization and homophily are relevant features.
References
Acemoglu, D., C. Garcıa-Jimeno, and J. A. Robinson (2015). State capacity and economic
development: A network approach. American Economic Review 105 (8), 2364–2409.
Aleman, E. and E. Calvo (2013). Explaining policy ties in presidential congresses: A network
analysis of bill initiation data. Political Studies 61 (2), 356–377.
Anderson, W. D., J. M. Box-Steffensmeier, and V. Sinclair-Chapman (2003). The keys to
legislative success in the us house of representatives. Legislative Studies Quarterly 28 (3),
357–386.
Baccara, M. and L. Yariv (2013). Homophily in peer groups. American Economic Journal:
Microeconomics 5 (3), 69–96.
Badev, A. (2017). Discrete games in endogenous networks: Theory and policy. Mimeo,
University of Pennsylvania.
Bala, V. and S. Goyal (2000). A noncooperative model of network formation. Economet-
rica 68 (5), 1181–1229.
Ballester, C., A. Calvo-Armengol, and Y. Zenou (2006). Who’s who in networks. wanted:
The key player. Econometrica 74 (5), 1403–1417.
Banerjee, A. V., A. G. Chandrasekhar, E. Duflo, and M. O. Jackson (2018). Changes in
social network structure in response to exposure to formal credit markets. SSRN paper
3245656 .
Barber, M. and N. McCarty (2015). Causes and consequences of polarization. Political
Negotiation: A Handbook 37, 39–43.
Bartels, L. M. (1993). Messages received: the political impact of media exposure. American
Political Science Review 87 (02), 267–285.
Battaglini, M. and E. Patacchini (2018). Influencing connected legislators. Journal of Po-
litical Economy 126 (6), 2277–2322.
Battaglini, M., E. Patacchini, and E. Rainone (2019). Endogenous social connections in
legislatures. Technical report, National Bureau of Economic Research.
Battaglini, M., V. L. Sciabolazza, and E. Patacchini (2018). Effectiveness of connected
legislators.
Baumann, L. (2017). A model of weighted network formation. SSRN Working Paper .
Bramoulle, Y., H. Djebbari, and B. Fortin (2009). Identification of peer effects through social
networks. Journal of Econometrics 150 (1), 41–55.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 43
Bratton, K. A. and S. M. Rouse (2011). Networks in the legislative arena: How group
Wang, Y. X. and Y. J. Zhang (2013). Nonnegative matrix factorization: A comprehensive
review. IEEE Transactions on Knowledge and Data Engineering 25 (6), 1336–1353.
Wilson, R. K. and C. D. Young (1997). Cosponsorship in the u. s. congress. Legislative
Studies Quarterly 22 (1), 25–43.
Zhang, Y., A. Friend, A. L. Traud, M. A. Porter, J. H. Fowler, and P. J. Mucha (2008).
Community structure in congressional cosponsorship networks. Physica A: Statistical Me-
chanics and its Applications 387 (7), 1705–1712.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 1
ONLINE APPENDIX
Endogenous Networks and Legislative Activity
Nathan Canen, Matthew O. Jackson and Francesco Trebbi
Appendix A. Proofs
Lemma A.1. ∑j 6=i
sisjmij(s) = si.
Proof.∑j 6=i
sisjmij(s) =∑
j 6=i,j∈P (i)
sisjmij(s) +∑
j 6=i,j /∈P (i)
sisjmij(s)
=∑
j 6=i,j∈P (i)
sisj
(p(i)
p(j)∑k∈P (i),k 6=i p(k)sk
+ (1− p(i)) (1− p(j))∑k 6=i(1− p(k))sk
)+
+∑
j 6=i,j /∈P (i)
sisj(1− p(i))(1− p(j))∑
k 6=i(1− p(k))sk
= si
(p(i)
(∑j 6=i,j∈P (i) p(j)sj∑k 6=i,k∈P (i) p(k)sk
)+ (1− p(i))
∑j 6=i,j∈P (i)(1− p(j))sj∑
k 6=i(1− p(k))sk
)+
+si(1− p(i))
(∑j 6=i,j /∈P (i)(1− p(j))sj∑
k 6=i(1− p(k))sk
)
= si
(p(i) + (1− p(i))
(∑j 6=i,j∈P (i)
∑j 6=i,j /∈P (i)(1− p(j))sj∑
k 6=i(1− p(k))sk
))
= si
(p(i) + (1− p(i))
(∑j 6=i(1− p(j))sj∑k 6=i(1− p(k))sk
))= si.
Proposition 2.1: The limit equilibrium is defined by equations (2.11)-(2.13).
Proof of Proposition 2.1. Recall that we have from equations (2.8) and (2.7), from the First
Order Conditions, that:
(A.1) c =αix∗i
+s∗2ix∗2i
,
and
(A.2)s∗ix∗i
= φi∑j 6=i
s∗jmij(s∗)x∗j .
2 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
We also use that:
x∗i = αiXP (i)(A.3)
s∗i = αiSP (i),(A.4)
for some XP (i), SP (i), which comes from the fact thats∗ix∗i
andx∗iαi
are the same for all agents
within a party. Let P (i) ∈ 1, 2 be arbitrary.
Using (A.3) in (2.8) implies:
c =αix∗i
+s∗2ix∗2i
=αi
αiXP (i)
+α2iS
2P (i)
α2iX
2P (i)
=1
XP (i)
+S2P (i)
X2P (i)
.
Multiplying both sides by X2P (i) yields:
cX2P (i) = XP (i) + S2
P (i),(A.5)
which is (2.13).
Let us now substitute (A.3) in (2.7):
αiSP (i)
αiXP (i)
= φP (i)
∑j 6=i
αjSP (j)mij(s∗)αjXP (j)
SP (i)
XP (i)
= φP (i)
∑j 6=i
α2jXP (j)SP (j)mij(s
∗)
= φP (i)
∑j 6=i
α2jXP (j)SP (j)
(p(i)
p(j)∑k∈P (i),k 6=i p(k)s∗k
+ (1− p(i)) (1− p(j))∑k 6=i(1− p(k))s∗k
)Ij∈P (i)
+φP (i)
∑j 6=i
α2jXP (j)SP (j)
((1− p(i)) (1− p(j))∑
k 6=i(1− p(k))s∗k
)Ij /∈P (i).
Note that for the first two terms, p(i) = p(j) because they are only summed when j ∈ P (i).
For the last, p(i) 6= p(j) as it is summed when j /∈ P (i).
Rewriting the above with this implies:
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 3
SP (i)
XP (i)
= φP (i)
∑j 6=i
α2jXP (i)SP (i)
(p(i)
p(i)∑k∈P (i),k 6=i p(i)s
∗k
+ (1− p(i)) (1− p(i))∑k 6=i(1− p(k))s∗k
)Ij∈P (i)
+φP (i)
∑j 6=i
α2jXP (j)SP (j)
((1− p(i)) (1− p(j))∑
k 6=i(1− p(k))s∗k
)Ij /∈P (i).
Using that s∗k = αkSP (k) leads to:
SP (i)
XP (i)
= φP (i)
∑j 6=i
α2jXP (i)SP (i)
(p(i)2
p(i)∑
k∈P (i),k 6=i αkSP (k)
+(1− p(i))2∑
k 6=i(1− p(k))αkSP (k)
)Ij∈P (i)
+φP (i)
∑j 6=i
α2jXP (j)SP (j)
((1− p(i))(1− p(j))∑k 6=i(1− p(k))αkSP (k)
)Ij /∈P (i).
Let us focus on the case of P (i) = 1, as the other case is symmetric.
S1
X1
= φ1
∑j 6=i
α2jX1S1
(p1∑
k∈P (i),k 6=i αkS1
+(1− p1)2∑
k 6=i(1− p(k))αkSP (k)
)Ij∈P (i)
+φ1
∑j 6=i
α2jX2S2
((1− p1)(1− p2)∑k 6=i(1− p(k))αkSP (k)
)Ij /∈P (i).
Finally, we use that:
∑k 6=i
(1− p(k))αkSP (k) =∑
k 6=i,k∈P (i)
(1− p(k))αkSP (k) +∑
k 6=i,k/∈P (i)
(1− p(k))αkSP (k)
=∑
k 6=i,k∈P (i)
(1− p1)αkS1 +∑
k 6=i,k/∈P (i)
(1− p2)αkS2
= (1− p1)S1
∑k 6=i,k∈P (i)
αk + (1− p2)S2
∑k 6=i,k/∈P (i)
αk
= (1− p1)S1A1 + (1− p2)S2A2.
To finalize the calculations, we use the simplification above for the denominators of the
second and third terms.
Note that only αj is now a function of the summand j itself, in the main expression. We
also note that we can now use the indicators of j ∈ P (i) for the first two terms, and j /∈ P (i)
of the last term, within sums. These observations lead to the final equation:
4 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
S1
X1
= φ1X1S1
∑j 6=i
α2j
(p1
S1
∑k∈P (i),k 6=i αk
+(1− p1)2
(1− p1)S1A1 + (1− p2)S2A2
)Ij∈P (i)
+X2S2φ1
∑j 6=i
α2j
((1− p1)(1− p2)
(1− p1)S1A1 + (1− p2)S2A2
)Ij /∈P (i)
= φ1X1S1
∑j 6=i,j∈P (i)
α2j
(p1
S1A1
+(1− p1)2
(1− p1)S1A1 + (1− p2)S2A2
)
+X2S2φ1
∑j 6=i,j /∈P (i)
α2j
((1− p1)(1− p2)
(1− p1)S1A1 + (1− p2)S2A2
)
= φ1X1S1B1
(p1
S1A1
+(1− p1)2
(1− p1)S1A1 + (1− p2)S2A2
)+X2S2φ1B2
((1− p1)(1− p2)
(1− p1)S1A1 + (1− p2)S2A2
)= φ1
(X1B1p1
A1
+X1S1B1(1− p1)2
(1− p1)S1A1 + (1− p2)S2A2
+X2S2B2(1− p1)(1− p2)
(1− p1)S1A1 + (1− p2)S2A2
)= φ1
(p1X1B1
A1
+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2X2S2
(1− p1)A1S1 + (1− p2)A2S2
).
Proof of Proposition 2.2. Recall that an interior equilibrium is a solution to (2.11) to (2.13).
So, rewriting these:
(A.6) S1 = X1φ1
(p1B1X1
A1
+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2
(1− p1)A1S1 + (1− p2)A2S2
).
(A.7) S2 = X2φ2
(p2B2X2
A2
+(1− p2)2B2S2X2 + (1− p1)(1− p2)B1S1X1
(1− p1)A1S1 + (1− p2)A2S2
).
(A.8) cX21 = X1 + S2
1 , cX22 = X2 + S2
2 .
Substituting (A.6) into (A.8) leads to
cX21 = X1 +X2
1φ21
(p1B1X1
A1
+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2
(1− p1)A1S1 + (1− p2)A2S2
)2
.
or
(A.9) cX1 = 1 +X1φ21
(p1B1X1
A1
+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2
(1− p1)A1S1 + (1− p2)A2S2
)2
.
There is a similar expression for S2, X2. Note that the right hand side of (A.9) lies above
the left hand side as we approach X1 = 0 (same for X2). To have an interior solution, we
need the right hand side to sometimes fall at or below the left hand side for positive X1.
Suppose that the equilibrium (when it exists) is such that X1 ≥ X2, and the other case is
analogous just reversing subscripts everywhere. Then the right hand side is less than what
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 5
we get by replacing X2 by X1, and so we want
(A.10) cX1 ≥ 1 +X31φ
21
(p1B1
A1
+(1− p1)2B1S1 + (1− p1)(1− p2)B2S2
(1− p1)A1S1 + (1− p2)A2S2
)2
.
for some interior X1. Rewriting
(A.11) cX1 ≥ 1 +X31φ
21
(p1B1
A1
+(1− p1)2B1 + (1− p1)(1− p2)B2
S2
S1
(1− p1)A1 + (1− p2)A2S2
S1
)2
.
The right hand side is maximized either at S2
S1= 0 or S2
S1=∞, and so it is sufficient to have
(A.12) cX1 ≥ 1 +X31φ
21
(p1B1
A1
+ (1− p1) max
[B1
A1
,B2
A2
])2
.
Let
D1 = p1B1
A1
+ (1− p1) max
[B1
A1
,B2
A2
]Then (A.12) can be rewritten as
(A.13) cX1 ≥ 1 +X31φ
21D
21.
for some positive X1. Note that
D1 ≤ D = max
[B1
A1
,B2
A2
]So, it is sufficient to have
(A.14) cX1 ≥ 1 +X31φ
21D
2.
for some positive X1.
It is necessary and sufficient to check that the left hand side and right hand side are tangent
at the point at which the slope of the right hand side is c. This happens at X1 =√
c3φ21D
2
and then the corresponding sufficient condition becomes:
(A.15) c
(c
3φ21D
2
)1/2
≥ 1 +
(c
3φ21D
2
)3/2
φ21D
2,
or
(A.16)2c3/2
3√
3≥ φ1D.
Having this hold also for the other case, leads to the claimed expression.
Proof of Proposition 2.3. Assume by way of contradiction that there is sFBi , xFBi first best
that is finite. Consider the allocation s′i, x′i = λsFBi , λxFBi , with λ > 1. The later
increases all politician’s utility by a cubic rate (from equation (2.3)), while the costs increase
quadratically. Hence, s′i, x′i is feasible and yields a higher utility to all agents, which is a
contradiction.
6 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Appendix B. Additional Aspects of the Theory
B.1. Best Response Dynamics. Best response dynamics are described as follows. Con-
sider starting at some vectors s0, x0. Then the best response dynamics are described by:
(B.1) sti = xt−1i φP (i)
∑j 6=i
mij(st−1)st−1
j xt−1j ,
and
(B.2) xti =αic
+ st−1i
φP (i)
c
∑j 6=i
mij(st−1)st−1
j xt−1j .
It follows that if s0 = 0, then mij(st−1) = 0 for all ij (recall Footnote 10) and we get
immediate convergence to sti = 0, xti = αic
for all t. Otherwise, st, xt will be positive for all t.
To see how these best response dynamics work for a special case, let us consider the
situation in which there is some S0, X0 such that s0i = αiS
0 and x0i = αiX
0 (which has to
eventually hold at any limit point).63
In that case, working with the limiting or continuum case, in which the matching function
is symmetric within a party, and presuming that St−1k > 0 for each party (which happens
after the first period if some s0j > 0 and otherwise the solution is already described above),
we end up with the following dynamics. For party k (letting k′ denote the other party):
(B.3) Stk = X t−1k φk
(mkk(S
t−1)BkSt−1k X t−1
k +mkk′(St−1)Bk′S
t−1k′ X
t−1k′
),
and
(B.4) X tk =
1
c+ St−1
k
φkc
(mkk(S
t−1)BkSt−1k X t−1
k +mkk′(St−1)Bk′S
t−1k′ X
t−1k′
).
where
mkk(St−1) =
pkSkAk
+(1− pk)2
(1− p1)S1A1 + (1− p2)S2A2
,
and
mkk′(St−1) =
(1− p1)(1− p2)
(1− p1)S1A1 + (1− p2)S2A2
.
B.2. Endogenous Partisanship. A natural extension of our model would be to endogenize
the pi’s. We comment here on potential directions and issues that arise.
First, it is easy to see that if one simply endogenized the pi’s within the current model
without introducing any costs of affecting pi, then the solutions would be corner solutions.
If a group can choose its pi without having any costs of selecting pi, then (generically in the
parameters) one of the two groups would want to be entirely partisan, since one of the two
groups would find interacting with itself more beneficial than interacting across the aisle.
Such a corner solution is clearly of little interest, and is incompatible with our empirical
estimates.
More generally, there are interactions, both within and across parties, that happen natu-
rally due to committee membership among other things and would be difficult to prevent,
and others that might be costly to encourage. This suggests that there would minimum
63This is also useful in determining the instability of equilibria.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 7
and maximum levels of partisanship that could be attained and also that one would need to
model a nonlinear cost of partisanship. Once one provided a nonlinear cost to capture the
high cost of going to either extreme of pi = 0 or pi = 1, one would end up with an interior
equilibrium. A challenge would be that this could be dependent upon the cost formulation,
and so one would need to work with a flexible enough cost function to allow the model to fit
the data.
Having three endogenous choices for each of the two parties - partisanship, social effort, and
legislative effort - would then end up producing a model for which analytic characterizations
of the equilibrium would no longer be possible, and for which the multiplicity of equilibria
would more difficult to ascertain. There would be two approaches. One would be to work
entirely with numerical simulations. Since the interest in endogenizing partisanship levels
would presumably be to understand how they interact with other variables and change
incentives, this would require a very rich and complex set of simulations, especially as they
would be sensitive to the choice of the cost function.
Another approach, and perhaps the most fruitful, would be to fix one of the other effort
variables and return to a model in which there are just two different action variables that
agents/parties are making. Given the importance of partisanship on the endogeneity of the
network, a starting point might be to fix the xi’s and then work with the other variables. This
could be an interesting approach for further research. We chose to work with endogenizing
the network and legislative effort, holding partisanship constant, as these seem to be the
first-order questions, but understanding partisanship is also a very interesting topic.
Appendix C. Additional Tables, Figures and Counterfactuals
In this section we present additional results referenced in the main text. We first discuss
two additional counterfactuals that illustrate the model’s comparative statics in Appendix
C.1: how changes in c and changes in γP (i) affect equilibrium effort choices. Section C.2 has
additional Tables and Figures references in the main text and in Appendix C.1.
C.1. Additional Counterfactuals.
C.1.1. Change in c. We now examine counterfactuals with respect to socialization costs c.
Specifically, by increasing c we increase the cost of legislative effort relative to social effort.
Table C.1 reports what would happen to the model equilibrium activity levels if c were
higher. This exercise makes evident the nonlinear effects emerging from social interactions
in our environment, drivers of the results in the previous sections.
We find a negative effect of increasing c by 1 or 2% on the likelihood of bill passage.
Moreover, this effect is quantitatively large, substantially more than linear, consistent with
the evidence so far pointing towards a substantial role for social complementarities across
legislators. There is an increase of bill success probability of approximately 15 percent on
average for Democrats, and 10 percent for Republicans, across all Congresses, vis-a-vis a
decrease of 1 percent in c. Similarly large effects are present for a 2 percent cut. These
magnitudes appear consistent with the overall importance of social interactions in legislative
activity reported in Section 5.
8 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Table C.1. Counterfactuals in c: Predicted (Proportional) Change in the(Mean) Probability of Bill Approval
Congress 105 106 107 108 109 110
Increase in 1% in c Democrats -0.0673 -0.156 -0.170 -0.149 -0.142 -0.156Republicans -0.0616 -0.118 -0.126 -0.114 -0.111 -0.123
Increase in 2% in c Democrats -0.128 -0.261 -0.278 -0.253 -0.243 -0.262Republicans -0.118 -0.204 -0.212 -0.198 -0.195 -0.212
The table presents the change in the average probability of bill approval under thecounterfactual, where the estimated cost c, is increased by either 1% or 2% (of theestimated value c). We do this by calculating the the implied optimal x∗i , s∗i fromProposition 2.1 under the appropriate value of c and calculate the probability of approvaldefined as 1
ζP (i)(s∗i )
2. We then find the percentual change over the predicted values under
Table 2.
The rationale for this effect is that an increase in c leads to a decrease in the choice of
legislative effort. Since the latter has a sufficiently strong complementarity with social effort,
socializing by the individual also decreases in this equilibrium (similarly to what happens
at the low effort equilibrium of Cabrales et al., 2011). As a result, returns to socializing
by others also decrease, who then further decrease their own legislative efforts until a new
equilibrium is reached. This feeds back and leads to a large (nonlinear) negative effect,
highlighting the importance of socializing in the approval of bills. These dynamics contrast
to an opposite positive feedback effect that can occur, for example, in the higher Pareto
equilibrium of Cabrales et al. (2011). In that case, an increase in c may lead to an increase
in bill approval. This is because, at the high equilibrium, a decrease in c makes socialization
more costly, which reduces the incentives for socialization. In turn, this reduces the returns
for xi due to the complementarity of effort choices.
C.1.2. Change in γ with φ constant. To conclude, let us recall that the shock to the bill
passage ε is assumed to be standard Pareto distributed with scale parameter γP (i) > 0,
hence a lower γ determines a lower median draw of the positive shock ε and a lower chance
of legislative success. We now investigate the effects of an institutional change in which γ
gets reduced. For each of the 105th-110th Congresses, Table C.2 shows what would happen
to equilibrium efforts if γ was reduced while keeping φ constant (noticing that ζ must adjust
inversely).
As the system of equations in Proposition 2.1 does not change - the equations depend
on φi directly - only the probability of approval is affected as per equation (2.3). Hence, γ
only changes the shape of the bill approval function. Quantitatively a decrease in γP (i) by
10 percent leads to a sizeable shift of the probability of approval curve to the left (which in
Table 1 are shown to vary from 9.57 to 12.85 percent). This is shown in Table C.2, which
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 9
Table C.2. Counterfactual in γ: Predicted Probability of Bill Approval
Congress 105 106 107 108 109 110
DemocratsDecrease in 10% in γDem 0.047 0.048 0.078 0.081 0.062 0.165
RepublicansDecrease in 10% in γRep 0.086 0.095 0.064 0.098 0.084 0.052
We calculate the average probability of bill approval by party, when γP (i) is decreasedby 10% (keeping both φDem, φRep constant). This is done by calculating the probabilityof bill approval, given by P (yi = 1) = 1
ζP (i)(s∗i )
2, with s∗i the solution to the non-
noisy equilibrium system from Proposition 2.1 (further details in Appendix G). We thendecrease γP (i) by 10%.
reports the average probabilities of bill approval under a smaller γP (i) for each party. As we
only change the values of γ in equation (4.4)-(4.5), the percentage change in the expected
probability of bill passage is linear, dropping by 10 percent as well.
C.2. Additional Tables and Figures. In this subsection, we include additional Tables
and Figures referenced in the main text and in Appendices A-C.
10 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Figure C.1. Examples of Alternative Congress 110 Networks
(a) More than 3 Directed Cosponsorships
(b) Committee Network
(c) Alumni Network
We show illustrations of alternative networks used in the literature that we use to compare our modelagainst. A link in the committee network exists if two legislators sit in one of the 7 main committeestogether (see the Data section). A link in the alumni network exists if two legislators attended thesame university within 8 years of one another. While in the empirical specification of equation (5.2)the cosponsorship network is taken as the amount of directed cosponsorships, we illustrate it hereby plotting the upper triangular matrix of directed cosponsorships, with a link formed if a legislatorcosponsors more than 3 bills by another.
Notes: Standard errors in parentheses. The table presents the results from the GMM estimation underthe second specification. That is, we replace the Grosewart measure by dummy variables for the mostimportant committees. The variable Leadership represents a dummy of whether the politician was theSpeaker, the Majority or Minority Leader, or the Majority or Minority Whip. Rep is a dummy variablefor belonging to the Republican Party. The estimates of φDem and φRep are their estimated sets. Theoptimal weighting matrix is used, and standard errors are estimated as discussed in Appendix F. Allother notes follow those in Table 2.
12 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Table C.4. Estimated and High Effort Equilibria
Congress 105 106 107 108 109 110
Effort Level in the Estimated Equilibrium:
SDem 0.922 1.506 1.443 1.403 1.412 1.484
SRep 0.724 0.916 0.808 0.831 0.881 0.988
XDem 4.478 5.680 5.111 5.058 5.206 5.484
XRep 4.231 4.773 4.196 4.235 4.426 4.722
Effort Level in the Higher Equilibrium:
SDem 3.972 2.296 2.058 2.196 2.302 2.285
SRep 2.075 1.200 1.009 1.110 1.217 1.324
XDem 9.816 7.081 6.138 6.389 6.729 6.882
XRep 6.326 5.188 4.460 4.613 4.903 5.226
We numerically assess whether the equilibrium we have estimated is the equilibrium with the highestvalues of social and legislative efforts (SDem, SRep, XDem, XRep). To do so, for each Congress we com-pute two (possibly) distinct solution to the system of equations in Proposition 2.1 under our estimatedparameters. First, we find the solutions under our estimated parameters, with starting values for effortlevels at those estimated in Table 2 (upper panel). Second, we search for a higher effort equilibrium(lower panel). This is done by searching for a solution to equations (2.11) - (2.13), while starting from avector with large values of effort relative to the estimated ones (namely, (SDem, SRep, XDem, XRep)+2).The table shows that there exists an equilibrium with higher levels of both social and legislative effortin all Congresses. These effort levels are higher than the equilibrium ones we estimated and that weobserve in the data. For Congress 107, the pointwise estimate of c has a numerical quirk which gener-ates instability in the algorithms to find zeros. Instead, we use a value within its confidence interval,c+0.007, presented above. This change does not significantly impact estimated equilibrium effort levels.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 13
Table C.5. Effects of Polarization on Equilibrium Effort Levels (and How toOverturn it), Additional Scenarios
Congress 105 106 107 108 109 110
Change in Effort Levels when pRep = 0.3,Relative to the Baseline in Table 5
SDem 0.004 0.251 0.242 0.312 0.232 0.086
SRep -0.003 0.035 0.022 0.043 0.027 0.000
XDem 0.006 0.430 0.393 0.507 0.380 0.143
XRep -0.003 0.048 0.027 0.054 0.036 0.000
Change in Effort Levels when pRep = 0.3,and ADem is 10% higher than estimated
SDem 0.154 0.202 0.130 0.198 0.238 0.221
SRep 0.073 0.124 0.088 0.107 0.130 0.151
XDem 0.214 0.344 0.209 0.318 0.389 0.370
XRep 0.088 0.174 0.111 0.138 0.176 0.219
Change in Effort Levels with Full Polarization,pDem = 1, Relative to the Baseline Estimates
SDem 0.040 0.142 0.095 0.144 0.165 0.169
SRep -0.025 -0.269 -0.295 -0.265 -0.259 -0.227
XDem 0.053 0.241 0.152 0.230 0.268 0.282
XRep -0.028 -0.331 -0.318 -0.298 -0.307 -0.297
We compute the effects of polarization on effort levels in each Congress, as in Table 5. We consider 3different scenarios and compare effort levels to those in the estimated equilibrium in Table C.4. In thefirst, we increase the partisan bias for only Republicans to 0.3 in all Congresses in the second panel. Inthe second, we show the equilibrium effort levels when partisan bias for Republicans is 0.3, but whenthe types of the Democrats (ADem =
∑i∈Dem αi) increases by 10% relative to estimated levels. In
the third, we show a case in which there is full polarization. We can see that Democrats may stillincrease effort levels with polarization, and will do so especially with higher types. Therefore, increasedpolarization means there is increased interaction within the Democratic party and with higher types,leading to higher spillovers that can compensate the lack of interactions with the additional politiciansin the opposition.
14 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Appendix D. Parametric Identification, with set identification of (φ1, φ2)
Recall from equation (2.16) that αi = ez′iβP (i) , and that we observe proxies for (s∗i , x
∗i ).
We also have that mij = (1−p1)(1−p2)(1−p1)A1S1+(1−p2)A2S2
if i, j are from opposing parties and mij =p(i)
AP (i)SP (i)+ (1−p(i))2
(1−p1)A1S1+(1−p2)A2S2if they are from the same. Let us denote mij = m12, if i, j
belong to different parties, mij = m11 if they both belong to party 1, and mij = m22 if they
both belong to party 2.
We now proceed with identification of the parameters of the model. Applying (2.14) and
(2.16) in (2.10), for an arbitrary politician i from party P (i) we obtain:
sieεi = ez
′iβP (i)SP (i), and(D.1)
log(si) = log(SP (i)) + z′iβP (i) − εi.(D.2)
Since Eεi = Eziεi = 0, we now have elementary moment conditions (like an OLS) to esti-
mate α.64 The moment conditions (D.1) for each party identify the respective party specific
parameters. To see this more clearly, one can use the moment equations just described to
get:
E(log(si)− log(SP (i))− z′iβP (i)
)= 0(D.3)
Ezi(log(si)− log(SP (i))− z′iβP (i)
)= 0.(D.4)
As long as E[1, zi][1, zi]′ is invertible, then βP (i) and log(SP (i)) are identified. βP (i)
is identified off the different zi for members of the same party. SP (i) is identified from the
average proxy for effort within a party (the constant in the regression within a party).65
Similarly for xi:
xievi = ez
′iβP (i)XP (i)
log(xi) = log(XP (i)) + z′iβP (i) − vi,(D.5)
which can be written as another moment condition in terms of the i.i.d. mean zero random
variable vi. XP (i) can be similarly identified for each party.
Since we know βP (i), SP (i), XP (i) for both parties, c is identified from equation (2.13),
where it is the only unknown. However, p1, p2 cannot be uniquely identified from the system
above. It is clear that p1, p2, φ1, φ2 show up only in the same 2 equations: (2.11) and (2.12).
64Identification with nonparametric α is proved in a following Appendix.65Implicitly, we require that zi does not include the constant, as that cannot be separately identified fromlog(SP (i)) without further assumptions. The average (log) socializing is party-specific.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 15
To identify φ1, φ2, we pursue a set identification approach. To do so, let us first notice that
equations (2.11) - (2.12) can be rewritten as66:
S1 = φ1X1 (B1S1X1m11 +B2S2X2m12)(D.6)
S2 = φ2X2 (B2S2X2m22 +B1S1X1m12) .(D.7)
To proceed with set identification of (φ1, φ2), we calculate all triples m11,m12,m22 that
are consistent with some p1, p2 ∈ [0, 1] × [0, 1]. Hence, any pair (φ1, φ2) that satisfies the
above equations for some (p1, p2) ∈ [0, 1]× [0, 1] is identified.67
A few comments about mij(s∗) are in order. Even if mij(s
∗) was recovered uniquely, that
would not guarantee a unique identification of p1, p2. There might be multiple p1, p2 that can
yield the same meeting probabilities (usually a continuum of them, governed by a hyperbolic
function). Second, mij(s∗) is not a parameter of interest for us, since it is a normalization
on the gij function that governs the probability of linking.
Having identified all other parameters of the model, we can now prove the identification
of the parameters γP (i), ζP (i). To do so, we use the data on bill passage.
D.0.1. Identifying the components of φi: γP (i) and ζP (i). We recall that φi =ζP (i)γP (i)
m, where
γ was the scale parameter of the distribution of εi (the shock on bill approval, such that the
same politician could have different bills passing or not), m was the institutional threshold
for approval of a bill (i.e. the minimum amount of support needed for approval), and ζ was
the return to the politician from the voters of having the bill approved. We further assumed
that m = 1.
If we want to identify the components of γP (i) and ζP (i), we can use the probability of bill
approval equation (2.3):
P (yi = 1) =γP (i)
m
∑j 6=i
gij(s∗)x∗ix
∗j .
This, in turn, can be rewritten using the First Order Condition for si (equation (A.2)) as:
66This follows from an alternative rewriting of the first order conditions presented in the model:
SP (i)
XP (i)= φP (i)
∑j 6=i
α2jSP (j)XP (j)mij(s
∗)
= φP (i)
∑j 6=i,j∈P (i)
α2jSP (j)XP (j)mij(s
∗) +∑j /∈P (i)
α2jSP (j)XP (j)mij(s
∗)
= φP (i)
SP (i)XP (i)mij(s∗)Ii,j∈P (i)
∑j 6=i,j∈P (i)
α2j + SP (−i)XP (−i)m12(s∗)
∑j /∈P (i)
α2j
.
67In Appendix H, we also provide a solution to point identify p1, p2, but that relies on additional momentconditions, coming from the second moments of the error terms of the proxy variables (si, xi). However,such a solution requires additional structure outside of the theoretical model.
16 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
P (yi = 1) =γP (i)
m
∑j 6=i
gij(s∗)x∗ix
∗j
=γP (i)
ms∗ix∗i
∑j 6=i
s∗jx∗jmij(s
∗)
=γP (i)
ms∗ix∗i
(s∗iφix∗i
)=
γP (i)
mφis∗2i
=1
ζP (i)
s∗2i .(D.8)
where P (yi = 1) is the probability that a bill from politician i is approved.
Since s∗i = sieεi , we can rewrite (D.8) as:
P (yi = 1) =1
ζP (i)
s2i e
2εi .(D.9)
Taking logs implies that:
logP (yi = 1) = log
(1
ζP (i)
)+ log(s2
i ) + 2εi.(D.10)
Since Eεi = 0, taking expectations on both sides means the only unknown is ζP (i), which
is identified when looking at si and yi within each party.
ζP (i) must also satisfy the following restriction, due to γP (i) being the scale parameter of
the Pareto distributed εi68:
mφis∗2i
≥ γP (i), ∀i
ζP (i) ≥ miniα2iS
2P (i)
Since (φ1, φ2) had been previously (set) identified, and m = 1, then γP (i) = φiζP (i)
is (set)
identified. This completes the identification of the model. Finally, we note the importance
of the measurement errors for identification of this model.
D.1. The need for exponential measurement errors even when α is parametric.
The exponential form for the measurement errors is very useful for two main reasons. First,
we bypass the truncation issue (having to guarantee that si ≥ 0 for any ε). Second, it is
very tractable (as seen in equation (D.1)).
68This, in turn, implies that the support for εi within each party is [γP (i),∞). The restriction holds empiri-cally.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 17
One could think that measurement errors would not have to show up on s∗i and/or x∗i .
However, measurement errors on s∗i , x∗i are needed even with α being parametric. To see
this, consider dividing equation (2.10) by (2.9):
(D.11)s∗ix∗i
=SP (i)
XP (i)
.
The α’s cancel out, and the model is rejected as (D.11) does not hold in the data (without
randomness). Hence, there must be an error term in both s∗i and x∗i .
Appendix E. Rewriting the Model in terms of Moment conditions over i
In this Section, we provide the derivation for transforming the model from the equations
in Proposition 2.1 to the moment equations described in Section 4. We first note that one
can stack equations (D.1) and (D.5) across both parties, to get:
where Ii∈P2 is an indicator of whether i is in party 2. We have simply introduced the
dummy variable for party 2 to stack up the equations. Just like in Ordinary Least Squares,
the parameter in front of the indicator recovers the difference of that variable across parties.
Note that the above equations can be rewritten as:
log(si) = ziβs − εi,(E.1)
log(xi) = ziβx − vi,(E.2)
where zi = [1, Ii∈P2 , z′i, z′iIi∈P2 ], βs = [log(S1), log(S2)−log(S1), β1, β2−β1], βx = [log(X1), log(X2)−
log(X1), β1, β2 − β1].
Recall that Eεi = Eziεi = 0, which is now rewritten together as Eziεi = 0. Similarly,
Evi = Ezivi = 0 is rewritten as Ezivi = 0.
We are now ready to rewrite the model in terms of moment conditions. Equations (E.1)
and (E.2) are rewritten in terms of moments as:
Ezi(log(si)− z′iβs) = 0,(E.3)
Ezi(log(xi)− z′iβx) = 0.(E.4)
Each of the above equations is k by 1, where k is the dimensionality of zi. Note that the
constant is included in zi.
Focusing on (2.13), for all i in Party 1, one way we can rewrite it is:
18 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
0 = c− 1
X1
− S21
X21
= c− 1
X1
− s2i
x2i
= c− 1
X1
− s2i e
2(εi−vi)
x2i
.
Simplifying and taking logs:
log(c− 1
X1
) = 2log
(eεi−vi
sixi
)= 2(log(si)− log(xi) + εi − vi).
where we used (2.9) and (2.10) and rewrote them with our proxies (si, xi).69
Since an analogous version holds for Party 2, one can rewrite the equation above across
parties as:
(E.5) E(
2(log(si)− log(xi))− log(c− 1
X1
)Ii∈P1 − log
(c− 1
X2
)Ii∈P2
)= 0,
where we have taken expectations on both sides over εi, vi, across all agents. This can be
rewritten as equation (4.3) by replacing Ii∈P1 = 1− Ii∈P2 . Finally, to rewrite equation (4.4),
we simply take expectations over both sides of equation (D.10).
Appendix F. Details on Estimation
We now provide further details on how the estimation procedure was implemented, in-
cluding the starting values for the numerical solution to the GMM optimizer and numerical
details on the computation of standard errors.
F.1. OLS and plug-in Approach as Starting Values for Optimization. For the start-
ing values for GMM optimization, we use a simple closed form estimate for the parameters of
interest. This new estimator is the OLS estimator for a subset of the parameters, and a plug-
in of those for the remaining parameters. Such an estimator is consistent, but inefficient.
Yet, it is a good starting value for the optimization procedure.
The inefficiency comes from an OLS approach to equations (4.8) and (4.9) neglecting
that βDem, βRep are the same across both equations. To note that we can find this OLS
estimator, (4.8) and (4.9) are the moment conditions associated with an OLS problem. The
OLS estimator is then given by:
βs = (Z ′Z)−1Z ′log(s),(F.1)
βx = (Z ′Z)−1Z ′log(x),(F.2)
69While we can replace X1 separately to obtain analogous equations, they will be linearly dependent, as weare already using (2.9) and (2.10) in the estimation and the previous equation has been introduced.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 19
where Z is an N by k matrix of z′i, log(s) and log(x) are the vectors of log(si) and log(xi)
respectively.
By our definition of βs, βx, such an estimator is consistent for log(S1), log(X1), log(S2), log(X2)
(from the first two parameters in each of βs, βx). Similarly, we recover consistent estimates
for β1, β2 in each equation. One replaces them into either equation (2.13) to recover c (that
was not present in any of the other equations).
F.2. Computation of the Optimal Weighting Matrix and of Standard Errors.
F.2.1. The Optimal Weighting Matrix and Computation of Estimates for the Standard Er-
rors. To compute the standard errors for our GMM estimates, we use a two step procedure
that is common in the literature. First, we estimate the model using an inefficient weighting
matrix W = I, with I the identity matrix, with starting values from the OLS and plug-in
approach (described above). Given these inefficient estimates, we then compute the optimal
weighting matrix for GMM, based on its asymptotic formula.
The optimal weighting matrix for GMM is defined as W = Ω−1, where
Ω = E(g(si, xi, yi, zi, θ)g(si, xi, yi, zi, θ)′). As it is well known, the asymptotic variance matrix
(of√n times) our parameters of interest is then given by (Γ′Ω−1Γ)−1, where Γ = E∂g(si,xi,θ)
∂θ′.
We compute Γ analytically, by taking derivatives of each moment equation in relation to
each parameter. We then replace the expectation by its empirical counterpart (the mean
across all politicians).
F.2.2. Finite Sample Corrections for the Standard Errors. In finite samples, Ω can be close
to singular. This appears to be the case in some of the specifications based on the set-up in
Appendix H, using second order moment restrictions. We provide corrections that improve
the finite sample performance, used only for those specifications in Appendix.
We implement the correction used in Cameron et al. (2011). This involves increasing the
standard errors in Ω by adding a small perturbation to its eigenvalues. This perturbation is
sufficient to remove singularity.
Such a procedure uses the spectral decomposition of Ω = DΛD′, where Λ is a diagonal
matrix of eigenvalues. We then add a small δΩ > 0 to the diagonal of Λ, therefore increasing
the eigenvalues of Ω. Since this procedure increases standard errors, the new standard errors
are still valid for our parameters.
In practice, we pick δΩ = 0.0001, and use it on the eigenvalues that are smaller than 10−7.
This is typically 1 or 2 of the eigenvalues of our estimated Ω. This correction is used for
both the calculation of the optimal weighting matrix, as well as for the standard errors of
the parameters.
Appendix G. Computation of Comparative Statics
Our estimates of SDem, SRep, XDem, XRep, are useful for our comparative statics and fitting
our model. However, those estimates are not necessarily the values that solve the equilibrium
system in Proposition 2.1. They are estimates that solve the system in expected value (the
moment conditions), as we only observe proxies for social and legislative effort.
20 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
To calculate the values that are consistent with our model, we solve the system in Proposi-
tion 2.1 using those values as starting points, under the estimated values of (c, φ1, φ2, βDem, βRep).
The solution are values S∗Dem, S∗Rep, X
∗Dem, X
∗Rep that, under the estimated parameters, solve
the original game.
For example, when we are interested in the changes in the probability of bill approval, we
use S∗P (i), αi and equation (4.4), i.e. P (yi = 1) = 1ζP (i)
(αiS∗P (i))
2.
Appendix H. Identification and Estimation using second moments of the
proxies of (s∗i , x∗i )
In this section, we impose additional structure on the measurement errors of the proxies,
denoted εi, vi, such that we can recover point identification of p1, p2, φ1, φ2. In the following
subsection, we then estimate this version of the model.
As before, let us introduce the measurement errors εi, vi, such that we observe proxies of
(s∗i , x∗i ) of the following form:
si = s∗i e−εi(H.1)
xi = x∗i e−vi .(H.2)
However, we now assume that:
εi = ωip(i) + ui(H.3)
vi = ωip(i) + ηi,(H.4)
with ui, ωi, ηi being i.i.d. across i and we have the (standard) assumptions, for all i: E(ui |zi, ωi, p(i)) = 0,Eu2
i = σ2u,E(ηi | zi, ωi, p(i)) = 0,Eη2
i = σ2η,E(ωi | zi, p(i)) = 0,Eω2
i = σ2ω.
These conditions imply the restrictions E(εi | zi) = E(vi | zi) = 0 used in the main section
of the paper.
The structure allows the measurement errors to vary by individual and by party. An
intuition for this is that, when partisanship increases, it is harder to observe the true so-
cial/legislative effort of a politician. This implies we get noisier proxies for s∗i , x∗i with more
partisanship. When pi = 0, we only have classical measurement error in the proxies. With
no partisanship, there are no differences in how well we observe socializing across politicians
and parties.
The identification arguments presented in the previous approach still hold, up to the
identification of (φ1, φ2). Let us continue from there.
Under the assumptions stated above, note that:
Eε2i = E(ωip(i) + ui)2
= E(ω2i p(i)
2 + u2i + 2uiωip(i))
= p(i)2σ2ω + σ2
u.(H.5)
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 21
Similarly,
Ev2i = p(i)2σ2
ω + σ2η(H.6)
E(εivi) = p(i)2σ2ω.(H.7)
We make the following normalization, such that all variances are now written in terms of
the variance of ω: σω = 1.70 In that case, from (H.7) we get that:
(H.8) p(i)2 = E(εivi).
Since we can take expectations over i ∈ P1 or i ∈ P2, we get identification of both p1 and
p2. Plugging in p(i) into (H.5) identifies σ2u. Plugging p(i) into (H.6) identifies σ2
η.
Note that the moment conditions given by equations (H.5), (H.6), (H.7) (for each party)
are of very simple form to use. Given (p1, p2), we can identify φ1 immediately from equation
(4.6). Analogously, φ2 follows from plugging in the previously identiied parameters in (4.7).
H.1. Estimation with Second Moment Conditions on the Proxies of s∗i , x∗i . To
estimate this version of the model, we can retain all moment conditions presented in the
main part of the text, while adding the moment equations (H.5), (H.6), (H.7), together with
a moment equation derived for each of φ1, φ2. To derive the latter, equations (2.11) - (2.12)
are rewritten as:
E (log(si)− log(xi)− log(φ1)−Ψ0) Ii∈P1 = 0.(H.9)
and
E (log(si)− log(xi)− log(φ2)−Ψ1) Ii∈P2 = 0.(H.10)
where
Ψ0 = log
(p1B1X1
A1
+(1− p1)2B1S1X1 + (1− p1)(1− p2)B2S2X2
(1− p1)A1S1 + (1− p2)A2S2
), and
Ψ1 = log
(p2B2X2
A2
+(1− p2)2B2S2X2 + (1− p1)(1− p2)B1S1X1
(1− p1)A1S1 + (1− p2)A2S2
).
Estimation then follows the routines described in the main part of the text, except we now
estimate φ1, φ2, p1, p2, σ2u, σ
2η.
H.2. Results under Restrictions on the Second Moments of the Proxies for (s∗i , x∗i ).
These are presented in Tables H.1 - H.2 below. We can also show that our results for the
counterfactuals by changing c and by changing α remain quantitatively and qualitatively
similar to those in Tables 6 and C.1. This shows the robustness of our results to different
parametrizations. However, due to space considerations, we do not present them here.
70p1, p2 are always multiplied by σω.
22 ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY
Table H.1. Results, Specification 1, second moments of the (s∗i , x∗i ) proxy
N 437 434 436 438 434 437Notes: Standard Errors in parentheses. The table presents the results from the GMM estimationunder second moment conditions on the proxies of (s∗i , x
∗i ). The optimal weighting matrix is used, and
standard errors are estimated as discussed in Appendix. All other notes follow those in Table 2.
ENDOGENOUS NETWORKS AND LEGISLATIVE ACTIVITY 23
Table H.2. Results, Specification 2, second moments of the (s∗i , x∗i ) proxy